# Smooth Selection for Infinite Sets

Fushuai Jiang  Department of Mathematics, UC Davis One Shields Ave Davis, CA
95616<EMAIL_ADDRESS>,  Garving K. Luli  Department of Mathematics,
UC Davis One Shields Ave Davis, CA 95616<EMAIL_ADDRESS>and  Kevin
O’Neill  Applied Mathematics Program, Yale University 51 Prospect Street New
Haven, CT 06511<EMAIL_ADDRESS>
###### Abstract.

Whitney’s extension problem asks the following: Given a compact set
$E\subset\mathbb{R}^{n}$ and a function $f:E\to\mathbb{R}$, how can we tell
whether there exists $F\in C^{m}(\mathbb{R}^{n})$ such that $F=f$ on $E$? A
2006 theorem of Charles Fefferman [6] answers this question in its full
generality.

In this paper, we establish a version of this theorem adapted for variants of
the Whitney extension problem, including nonnegative extensions and the smooth
selection problems. Among other things, we generalize the Finiteness Principle
for smooth selection by Fefferman-Israel-Luli [9] to the setting of infinite
sets.

Our main result is stated in terms of the iterated Glaeser refinement of a
bundle formed by taking potential Taylor polynomials at each point of $E$. In
particular, we show that such bundles (and any bundles with closed, convex
fibers) stabilize after a bounded number of Glaeser refinements, thus
strengthening the previous results of Glaeser, Bierstone-Milman-Pawłucki, and
Fefferman which only hold for bundles with affine fibers.

##  1\. Introduction

Let $m,n,d$ be positive integers. We write
$C^{m}(\mathbb{R}^{n},\mathbb{R}^{d})$ to denote the space of all functions
$\vec{F}:=(F_{1},\cdots,F_{d}):\mathbb{R}^{n}\rightarrow\mathbb{R}^{d}$ whose
derivatives
$\partial^{\alpha}\vec{F}=(\partial^{\alpha}F_{1},\cdots,\partial^{\alpha}F_{d})$
(for all $|\alpha|\leq m$) are continuous and bounded on $\mathbb{R}^{n}$. We
equip $C^{m}(\mathbb{R}^{n},\mathbb{R}^{d})$ with the usual norm

$\|{\vec{F}}\|_{C^{m}(\mathbb{R}^{n},\mathbb{R}^{d})}:=\max_{\begin{subarray}{c}|\alpha|\leq
m,\\\ 1\leq j\leq
d\end{subarray}}\sup_{x\in\mathbb{R}^{n}}|\partial^{\alpha}F_{j}(x)|$

and seminorm

$\|{\vec{F}}\|_{\dot{C}^{m}(\mathbb{R}^{n},\mathbb{R}^{d})}:=\max_{\begin{subarray}{c}|\alpha|=m,\\\
1\leq j\leq
d\end{subarray}}\sup_{x\in\mathbb{R}^{n}}|\partial^{\alpha}F_{j}(x)|.$

When $d=1$, we write $C^{m}(\mathbb{R}^{n})$ instead of
$C^{m}(\mathbb{R}^{n},\mathbb{R}^{d})$.

Whitney’s extension problem [33] asks, given a compact set
$E\subset\mathbb{R}^{n}$ and a function $f:E\to\mathbb{R}$, how can we tell
whether there exists $F\in C^{m}(\mathbb{R}^{n})$ such that $F|_{E}=f$? The
following nonnegative variation of Whitney’s extension problem has attracted
some recent attention (see [9, 10, 21, 20, 22]).

######  Problem 1.1 (Nonnegative extension).

Given integers $m\geq 0,n\geq 1$, an arbitrary subset
$E\subset\mathbb{R}^{n}$, $f:E\rightarrow[0,\infty)$, how can we tell if there
exists $F\in C^{m}(\mathbb{R}^{n})$ such that $F\geq 0$ and $F|_{E}=f$?

Simple examples, such as $E=\\{0\\}\cup\\{\frac{1}{n}:n\in\mathbb{N}\\}$ with
$f(x)=x$ and $m\geq 1$, demonstrate that the existence of a nonnegative
$C^{m}$ extension does not follow merely from nonnegative $f$ and the
existence of a (not necessarily nonnegative) $C^{m}$ extension. Thus, a
solution to Problem 1.1 must address an obstacle absent from the classical
Whitney extension problem. Naturally, this obstacle extends to the further
generalizations explored in this paper.

Problem 1.1 can be viewed as a special case of the following generalized
problem (see [11] for recent progress on the analogous problem for finite
sets):

######  Problem 1.2 (Extension with restricted range).

Given extended real numbers $\lambda_{1},\lambda_{2}$ with
$-\infty\leq\lambda_{1}\leq\lambda_{2}\leq\infty$, $E\subset\mathbb{R}^{n}$,
and $f:E\to[\lambda_{1},\lambda_{2}]\cap\mathbb{R}$, how can we tell if there
exists $F\in C^{m}(\mathbb{R}^{n})$ such that $\lambda_{1}\leq
F\leq\lambda_{2}$ and $F|_{E}=f$?

Our solution to Problems 1.1 and 1.2 will result from the solution to an even
more generalized problem. In order to present our solution to this problem, we
need to introduce some background definitions and recall some facts.

If $\vec{F}\in C^{m}(\mathbb{R}^{n},\mathbb{R}^{d}),$ we write
$J_{x}\vec{F}=(J_{x}F_{1},\cdots,J_{x}F_{d})$ (the “$m$-th jet”) to denote the
$m$-th degree Taylor polynomial of $\vec{F}$ at $x$. Let $\mathcal{P}$ denote
the vector space of polynomials in $n$ real variables and of degree at most
$M$. Given $x\in\mathbb{R}^{n}$, we give $\mathcal{P}$ the structure of a
ring, denoted $\mathcal{R}_{x}$, by defining multiplication $\odot_{x}$ as
$R\odot_{x}R^{\prime}=J_{x}(RR^{\prime})$. We write $\vec{\mathcal{R}}_{x}$
for the $\mathcal{R}_{x}$-module of $m$-jets of functions $\vec{F}\in
C^{m}(\mathbb{R}^{n},\mathbb{R}^{d})$ at $x\in\mathbb{R}^{n}$. As a vector
space, $\vec{\mathcal{R}}_{x}$ can be identified with
$\vec{\mathcal{P}}:=\underbrace{\mathcal{P}\oplus\cdots\oplus\mathcal{P}}_{d\text{
copies}}$, the space of $d$-tuples of real $m$-th degree polynomials on
$\mathbb{R}^{n}$. The $\mathcal{R}_{x}$-multiplication on
$\vec{\mathcal{R}}_{x}$ is given by
$R\odot_{x}\vec{P}=\left(R\odot_{x}P_{1},\cdots,R\odot_{x}P_{d}\right)$, where
$R\in\mathcal{R}_{x}$ and $\vec{P}=(P_{1},...,P_{d})$.

We write $c(m,n,d),C(m,n,d),$ etc., to denote constants that depend on
$m,n,d$; they may denote different constants in different appearances.

If $S$ is a finite set, we write $\\#(S)$ to denote the number of elements in
$S$.

A problem that is closely related to Problems 1.1 and 1.2 is the following

######  Problem 1.3 (Smooth Selection).

Let $E\subset\mathbb{R}^{n}$ be an arbitrary set. For each $x\in E$, let
$\mathcal{K}(x)\subset\mathbb{R}^{d}$ be closed and convex. How can we tell
whether there exists a function $\vec{F}\in
C^{m}(\mathbb{R}^{n},\mathbb{R}^{d})$ such that $\vec{F}(x)\in\mathcal{K}(x)$
for all $x\in E$?

The pioneer papers [28, 9, 5] studied the analogous version of Problem 1.3 for
finite sets $E$. In particular, for finite sets $E\subset\mathbb{R}^{n}$, the
authors in [9] gave an answer to the finite set version of the problem by
means of proving a finiteness principle. See also [16, 30, 29, 25, 31, 26, 27]
for related Lipschitz selection problems and [10, 21, 22, 20] for related
problems of nonnegative interpolation.

The $C^{1}$ case of nonnegative extension (Problem 1.1) was addressed in [19],
though little other work has directly addressed the infinite set versions of
the constrained extension problems addressed here.

In this paper, we modify the approach of [6] to answer Problem 1.3 for
infinite sets $E\subset\mathbb{R}^{n}$.

Before we present our solution to the above problems, we would like to point
out another closely related problem for which our result can provide an
answer:

######  Problem 1.4 (Generalized Brenner-Epstein-Hochster-Kollár Problem).

Suppose we are given real-valued functions $\phi_{1},\cdots,\phi_{d}$, an
$\mathbb{R}^{s}$-valued function $\phi$ on $\mathbb{R}^{n}$, and closed convex
subsets $K_{1}(x),\cdots,K_{d}(x)\subset\mathbb{R}^{s}$ for each $x\in
E\subset\mathbb{R}^{n}$. How can we decide whether there exist
$f_{1},\cdots,f_{d}\in C^{m}(\mathbb{R}^{n},\mathbb{R}^{s})$ such that

(1.1) $\sum_{i=1}^{d}\phi_{i}f_{i}\leq\phi\text{ on }E$

and

(1.2) $f_{i}(x)\in K_{i}(x),\text{ for }1\leq i\leq d,x\in E?$

The pioneer papers [2, 14, 15] studied Problem 1.4 where the inequalities in
(1.1) are replaced by equations and there are no convex constraints (1.2).
Note that the inequalities in (1.1) may be used to restrict the output of the
$f_{i}$ to convex sets depending on $x$, provided said convex sets are the
intersection of a bounded number of half-spaces.

In order to present our solutions to Problems 1.2 (a fortiori 1.1), 1.3, and
1.4, we introduce some basic definitions.

Let $E\subset\mathbb{R}^{n}$ be a compact set. A bundle over $E$ is a family
$\mathcal{H}=(H(x))_{x\in E}$ of (possibly empty) subsets
$H(x)\subset\vec{\mathcal{P}}$, parameterized by the points $x\in E$. We refer
to each $H(x)$, $x\in E$, as a fiber. A $C^{m}$ section (or “section” for
short) of a bundle $\mathcal{H}=(H(x))_{x\in E}$ is a $C^{m}$ function
$\vec{F}:\mathbb{R}^{n}\to\mathbb{R}^{d}$ such that $J_{x}\vec{F}\in H(x)$ for
each $x\in E$. The problem of finding a $C^{m}$ extension satisfying certain
conditions reduces to the problem of finding a section of a given bundle (see
below).

We say a bundle $(H(x))_{x\in E}$ is convex if $H(x)$ is convex for each $x\in
E$. (We note that prior literature has required each fiber of a bundle to be
an affine space; we drop the requirement to handle nonlinear problems.)

Fix integers $m\geq 0,n,d\geq 1$. Let $(H(x))_{x\in E}$ be a bundle and let
${k^{\sharp}}$ be a positive integer depending only on $m,n,d$. For each
$x_{0}\in E$, we define the Glaeser refinement of $H(x_{0})$, denoted by
$\tilde{H}(x_{0})$, according to the following rule:

  * (1.3)

Let $\vec{P}_{0}\in\vec{\mathcal{P}}$. We say that
$\vec{P}_{0}\in\tilde{H}(x_{0})$ if and only if given $\epsilon>0$, there
exists $\delta>0$ such that, for any $x_{1},\cdots,x_{{k^{\sharp}}}\in E\cap
B(x_{0},\delta)$, there exist
$\vec{P}_{1},\cdots,\vec{P}_{{k^{\sharp}}}\in\vec{\mathcal{P}}$, with
$\vec{P}_{j}\in H(x_{j})$ for $j=0,1,\cdots,{k^{\sharp}}$ and
$\left|\partial^{\alpha}(\vec{P}_{i}-\vec{P}_{j})(x_{j})\right|\leq\epsilon\left|x_{i}-x_{j}\right|^{{m-\left|\alpha\right|}}$
for $\left|\alpha\right|\leq m$, $0\leq i,j\leq{k^{\sharp}},1\leq l\leq d$.

The Glaeser refinement is used, essentially, to toss polynomials which cannot
satisfy the conditions of Taylor’s theorem with the polynomials remaining at
nearby fibers.

We say that a bundle $(H(x))_{x\in E}$ is Glaeser stable if $H(x)$ is its own
Glaeser refinement for each $x\in E$.

We will often produce repeated Glaeser refinements of the same bundle, in
which case $(H_{0}(x))_{x\in E}$ will be used to denote the original bundle
and $(H_{k+1}(x))_{x\in E}$ will be used to denote $(\tilde{H_{k}}(x))_{x\in
E}$ for $k\in\mathbb{N}$.

The Glaeser refinement is a procedure first suggested in 1958 by G.
Glaeser[18] for determining whether a function $f$ defined on an arbitrary
subset $E\subset\mathbb{R}^{n}$ can be extended to a $C^{1}(\mathbb{R}^{n})$
function. In their studies of extension problems in $C^{m}(\mathbb{R}^{n})$
for subanalytic sets, Bierstone, Milman and Pawłucki [1] introduced the notion
of iterated paratangent bundles, which is analogous to the Glaeser refinement.
Our version of the Glaeser refinement is a vector-valued version of the one
used by C. Fefferman for extending a function $f$ defined on an arbitrary
subset $E\subset\mathbb{R}^{n}$ to a $C^{m}(\mathbb{R}^{n})$ function. We also
apply our Glaeser refinement to a more general class of bundles than in [6].

Here is an outline for the rest of the introduction. First, we briefly recall
Fefferman’s solution [6] to the Whitney’s extension problem for $C^{m}$ and
then we will point out the major hurdles in adapting the machinery to prove
our main theorems (see Theorems 1.5, 1.6 below). We will then state our main
results and explain our proof in broad strokes.

Given a function $f:E\rightarrow\mathbb{R}^{n}$ on an arbitrary subset
$E\subset\mathbb{R}^{n}$, to determine whether $f$ can be extended to a
function $F\in C^{m}(\mathbb{R}^{n})$ – This is called the “Whitney Extension
Problem for $C^{m}$”– C. Fefferman [6] associated an affine subspace
$H(x)\subset\mathcal{P}$ for each $x\in E$, with the following crucial
property:

  * •

If $F\in C^{m}\left(\mathbb{R}^{n}\right)$ and $F=f$ on $E,$ then
$J_{x}\left(F\right)\in H(x)$ for all $x\in E$.

Evidently, if $H(x)=\emptyset$ for any $x\in E$, then $f$ cannot be extended
to a $C^{m}$ function $F$. Fefferman’s solution [6] to the Whitney’s extension
problem for $C^{m}$ can be summarized as follows

  1. (1)

We start with the trivial holding space
$H_{0}(x)=\left\\{P\in\mathcal{P}:P(x)=f(x)\right\\}$ for all $x\in E$.

  2. (2)

We produce a list of holding spaces $H_{0}(x)\supset H_{1}(x)\supset
H_{2}(x)\supset\cdots$ by repeatedly taking the Glaeser refinement (see
Definition 1).

  3. (3)

After at most $L=2\dim\mathcal{P}+1$, we have $H_{L}(x)=H_{l}(x)$ for all
$l\geq L$ and all $x\in E$. (In other words, $(H_{L}(x))_{x\in E}$ is Glaeser
stable.)

  4. (4)

$f$ extends to a $C^{m}$ function on $\mathbb{R}^{n}$ if and only if
$H_{L}(x)$ is non-empty for all $x\in E$.

Fefferman’s proof of the solution relies on the affine space structure of the
$H(x)$ (and breaks down for more general sets) in three crucial ways:

  1. (1)

The proof that iterated Glaeser refinements terminate in a Glaeser stable
bundle after a finite number of steps (Step 3) in [6] (adapted from [1])
relies on the affine space structure of $H_{l}$ and an induction on the
dimensions. It uses the fact that if $W\subset V$ is a subspace with $\dim
W=\dim V$, then $W=V$, which is false for the natural notion of dimension for
convex sets.

  2. (2)

The proof that the $\delta$ in the definition of Glaeser stability may be
chosen uniformly in $x$ and $P$ (with some adjustment) relies on the finite
representation of $H(x)$ via a linear basis. General convex sets may not be
represented in terms of a finite basis.

  3. (3)

The set $E$ is partitioned according to the dimension of the $H(x)$ as affine
spaces. If one does this by dimension of a convex set, it must be checked this
dimension does not change under natural set operations.

Our circumstances do not guarantee the presence of affine spaces. For
instance, considering Problem 1.1 we see that $x^{2}$ and $2x^{2}$ are
potential Taylor polynomials at $x=0$, while $-x^{2}$ is not.

To establish the termination of iterated Glaeser refinements of bundles
$(H(x))_{x\in E}$ whose fibers $H(x)$ are allowed to be general convex sets,
we recognize the following: The Glaeser refinement depends on a local
procedure (see Definition 1). More precisely, in order to check if
$\vec{P}_{0}\in\tilde{H}(x_{0})$, we need to consider only polynomials
$\vec{P}_{i}$ that are close to $\vec{P}_{0}$. Furthermore, convex subsets of
$\mathbb{R}^{n}$ have a notion of dimension which allows us to view them as
affine spaces locally. To formalize this idea, we introduce the local
dimension in Definition 3.1.

To find uniform $\delta$ as dictated by (2) above, we pass from the infinite
to the finite via compactness of $E$ and the $H(x)$, truncating the latter as
necessary. An open cover of neighborhoods in $E\times\mathcal{P}$ must be
carefully constructed so that each of these neighborhoods allows for uniform
choice of $\delta$ (for given $\epsilon$) and for the necessary quantitative
analysis to be conducted. This delicate construction is done in Section 9.
Sorting the elements $x$ of $E$ by $\dim H(x)$ is initially easy; any convex
set naturally caries a dimension. However, we must show that the truncations
described above do not change the dimension and ruin the partition. This is
done in Section 7.

To state our main theorems, we need to introduce a few terms.

Let $E\subset\mathbb{R}^{n}$ be an arbitrary set. For each $x\in E$ and $M\geq
0$, let $\Gamma\left(x,M\right)\subset\mathcal{\vec{P}}$ be a (possibly empty)
convex set. We say $\Gamma=(\Gamma\left(x,M\right))_{x\in E,M\geq 0}$ is a
shape field if for all $x\in E$ and $0\leq M^{\prime}\leq M<\infty$, we have
$\Gamma(x,M^{\prime})\subset\Gamma\left(x,M\right)$. We say a shape field
$\Gamma=(\Gamma\left(x,M\right))_{x\in E,M\geq 0}$ is closed if $\Gamma(x,M)$
is closed for each $x\in E$ and $M\geq 0$.

Any shape field $(\Gamma(x,M))_{x\in E,M\geq 0}$ on a compact set $E$ gives
rise to a convex bundle $(\Gamma(x))_{x\in E}$ if we set

$\Gamma(x)=\bigcup_{M\geq 0}\Gamma(x,M)\text{ for each }x\in E.$

We may then define the Glaeser refinement of a shape field
$(\Gamma(x,M))_{x\in E,M\geq 0}$ to be $(\tilde{\Gamma}(x,M))_{x\in E,M\geq
0}$, where $\tilde{\Gamma}(x,M)=\tilde{\Gamma}(x)\cap\Gamma(x,M)$ with
$\tilde{\Gamma}(x)$ being the Glaeser refinement of $\Gamma(x)$.

We will also define what it means for a shape field to be regular (see
Definition 2.1) or $(C,\delta_{\text{max}})$-convex for
$\delta_{\text{max}}>0$ (see Definition 2.2), though in the interest of
postponing technicalities this will be presented in Section 2. For now, we
simply note that regularity ensures $\Gamma(x,M)$ “varies continuously” in $x$
and $M$, while $(C,\delta_{\text{max}})$ convexity allows us to combine
functions via partition of unity as in [9]. Both properties will be satisfied
by reasonable examples of shape fields, in particular, the examples arising
from Problems 1.1-1.4.

We are now ready to state our main theorems.

######  Theorem 1.5 (Qualitative Main Theorem).

Let $m\geq 0,n,d\geq 1$ be integers. There exists
$k^{\sharp}=k^{\sharp}(m,n,d)$ in ((1.3)) such that the following holds.

Let $E\subset\mathbb{R}^{n}$ be a closed set. Let $(\Gamma(x))_{x\in E}$ be a
bundle arising from a closed regular $(C,1)$-convex shape field. Then the
iterated Glaeser refinement of $(\Gamma(x))_{x\in E}$ terminates in a Glaeser
stable bundle $(\Gamma^{*}(x))_{x\in E}$ in at most $2\dim\vec{\mathcal{P}}+1$
steps and $(\Gamma(x))_{x\in E}$ has a section if and if every fiber of
$(\Gamma^{*}(x))_{x\in E}$ is nonempty.

We can further strengthen Theorem 1.5 to have better control on the
derivatives. In fact, this step is required for our proof strategy. We need
the following definition for this purpose.

Given a shape field $\Gamma=(\Gamma(x,M))_{x\in E,M\geq 0}$ with nonempty
fibers, we define

(1.5) $\|\Gamma\|:=\inf\left\\{M\geq 0:\,\begin{matrix}[l]\text{For all
$x_{1},\cdots,x_{k^{\sharp}}\in E$, there exist }\\\
\text{$\vec{P}_{i}\in\Gamma(x_{i},M)$ for $1\leq i\leq{k^{\sharp}}$ such that
}\\\ \text{$\left|\partial^{\alpha}(\vec{P}_{i}-\vec{P}_{j})(x_{i})\right|\leq
M\left|x_{i}-x_{j}\right|^{m-\left|\alpha\right|}$, $1\leq
i,j\leq{k^{\sharp}}$.}\end{matrix}\right\\}\,.$

The finiteness of $\|\Gamma\|$ will be established in Lemma 6.1.
(Specifically, we prove this for the scalar-valued case in Section 6 and
reduce to this case in Section 4.)

######  Theorem 1.6 (Quantitative Main Theorem).

Let $m\geq 0,n,d\geq 1$ be integers. There exists
$k^{\sharp}=k^{\sharp}(m,n,d)$ in ((1.3)) and (1.5) such that the following
holds:

Fix $C_{w}>0$. Let $Q_{0}\subset\mathcal{\mathbb{R}}^{n}$ be a cube of length
$3$ and $E\subset Q_{0}$ be compact. Let $(\Gamma(x,M))_{x\in E,M\geq 0}$ be a
closed regular $(C_{w},1)$-convex shape field. Write

(1.6) $\Gamma(x)=\bigcup_{M\geq 0}\Gamma(x,M).$

Then the following hold.

  1. (1)

The repeated Glaeser refinement of $(\Gamma(x))_{x\in E}$ terminates in a
Glaeser stable bundle $(\Gamma^{\ast}(x))_{x\in E}$ after at most $2D+1$
steps, where $D=D(m,n,d)=\dim\vec{\mathcal{P}}$.

  2. (2)

If $\Gamma^{\ast}(x)$ is nonempty for all $x\in E$, then there exists
$\vec{F}\in C^{m}(\mathcal{\mathbb{R}}^{n},\mathcal{\mathbb{R}}^{d})$ such
that

(1.7)
$\|\vec{F}\|_{\dot{C}^{m}(\mathcal{\mathbb{R}}^{n},\mathcal{\mathbb{R}}^{d})}\leq
C\|\Gamma^{\ast}\|$

and

(1.8) $J_{x}\vec{F}\in\Gamma(x,C\|\Gamma^{\ast}\|)\text{ for all }x\in E.$

Here the constant $C$ depends only on $m,n,d,C_{w}$.

In Section 2, we will apply Theorem 1.6 to present our solutions to Problems
1.3, 1.2, and 1.4. In Section 3, we will prove the first part of Theorem 1.6,
using the aforementioned concept of local dimension. Section 4 contains a
reduction to the case $d=1$, using the gradient trick as in [15].

In Sections 5 through 10, we prove the second half of Theorem 1.6. We show
that for given $\epsilon>0$, we may find a uniform $\delta$ in the statement
that $(\Gamma^{*}(x))_{x\in E}$ is Glaeser stable and use this to construct a
modulus of continuity for which we may apply the $C^{m,\omega}$ shape fields
finiteness principle from [9]. The proof that $\delta$ may be taken uniformly
will rely on the dimension of $\Gamma(x)$ being the same for all $x\in E$,
forcing us to decompose $E$ into smaller sets (strata) for which this property
holds. The solutions are then patched together via a partition of unity. The
process is formally conducted via induction on the number of strata.

Section 11 is not part of the proof of Theorem 1.6; however, it will provide a
drastic decrease in the value of ${k^{\sharp}}$ from that given in the proof
of the theorem (greater than $(\dim\vec{\mathcal{P}}+1)^{3\cdot
2^{\dim\vec{\mathcal{P}}}}$) to ${k^{\sharp}}=2^{\dim\vec{P}}$.

This is an overly simplified version of our long story. The details will be
presented in the sections below.

This paper is part of a literature on extension and interpolation, going back
to the seminal works of H. Whitney [32, 33, 34]. We refer the interested
readers to [12, 13, 7, 21, 9, 10, 8, 17] and references therein for the
history and related problems.

######  Acknowledgment.

We are grateful to Charles Fefferman and Pavel Shvartsman for valuable
discussions and suggestions.

The first author is supported by the UC Davis Summer Graduate Student
Researcher Award and the Alice Leung Scholarship in Mathematics. The second
author is supported by NSF Grant DMS-1554733 and the UC Davis Chancellor’s
Chancellor’s Fellowship.

##  2\. Applications

In this section, we explain how to use Theorem 1.5 to answer Problems 1.3,
1.2, and 1.4.

By specializing to $d=1$, $E=\mathbb{R}^{n}$, and

$\mathcal{K}(x)=\begin{cases}\left\\{f(x)\right\\}\text{ for }x\in E_{0}\\\
[\lambda_{1},\lambda_{2}]\cap\mathbb{R}\text{ for }x\in E\setminus
E_{0}\end{cases}\,,$

we see that Problem 1.3 encompasses Problem 1.2.

By specializing to $D=ds$, $E=E_{0}$ and

$\mathcal{K}(x)=\left\\{(y_{1},\cdots,y_{d})\in\prod_{i=1}^{d}K_{i}(x):y_{i}\in\mathbb{R}^{s}\text{
for }i=1,\cdots,d,\text{ and
}\sum_{i=1}^{d}\phi_{i}(x)y_{i}\leq\phi(x)\right\\},$

we see that Problem 1.3 encompasses Problem 1.4.

Thus, to solve Problems 1.2 and 1.4, it suffices to solve Problem 1.3.

In order to show Theorem 1.5 applies to Problem 1.3, we must provide the
definitions missing from the introduction.

######  Definition 2.1.

Given a compact set $E\subset\mathbb{R}^{n}$, we say a shape field
$(\Gamma(x,M))_{x\in E,M\geq 0}$, with $\Gamma(x)=\bigcup_{M\geq
0}\Gamma(x,M)$, is regular if the following hold.

  * (2.1)

Given $M\geq 0,x\in E$,

$\Gamma(x,M)\subset\\{\vec{P}\in\mathcal{P}:|\partial^{\alpha}\vec{P}(x)|\leq
M\text{ for }|\alpha|\leq m\\}.$

  * (2.3)

Given $\epsilon>0,M\geq 0$, there exists $\delta>0$ such that for any
$x,x^{\prime}\in E$ with $\left|x-x^{\prime}\right|\leq\delta$, if
$\vec{P}\in\Gamma(x,M)$ and $\vec{P}^{\prime}\in\Gamma(x^{\prime})$ with

$\left|\partial^{\alpha}(\vec{P}-\vec{P}^{\prime})(x)\right|,\left|\partial^{\alpha}(\vec{P}-\vec{P}^{\prime})(x^{\prime})\right|\leq\delta\left|x-x^{\prime}\right|^{m-|\alpha|}\text{
for }|\alpha|\leq m,$

then $\vec{P}^{\prime}\in\Gamma(x^{\prime},M+\epsilon)$.

  * (2.5)

Given $\epsilon>0,M\geq 0$, there exists $\delta>0$ such that if
$\vec{P}\in\Gamma(x,M)$, then for any $\vec{P}^{\prime}\in\Gamma(x)$
satisfying

$|\partial^{\alpha}(\vec{P}-\vec{P}^{\prime})(x)|<\delta\text{ for
}|\alpha|\leq m$

implies

$\vec{P}^{\prime}\in\Gamma(x,M+\epsilon).$

Note that the above conditions say nothing about the content of $\Gamma(x)$,
rather the values of $M$ for which a given $\vec{P}\in\Gamma(x)$ lies in
$\Gamma(x,M)$.

######  Definition 2.2.

Let $\Gamma=\left(\Gamma\left(x,M\right)\right)_{x\in E,M\geq 0}$ be a shape
field. Let $C_{w}>0$ and $\delta_{\max}\in(0,1]$. We say that $\Gamma$ is
$(C_{w},\delta_{\max})$-convex, if the following condition holds.

Let $0<\delta\leq\delta_{\max}$, $x\in E,M\geq 0$,
$\vec{P}_{1},\vec{P}_{2}\in\mathcal{\vec{P}}$, $Q_{1},Q_{2}\in\mathcal{P}$.
Assume that

(2.7) $\displaystyle\vec{P}_{1},\vec{P}_{2}\in\Gamma\left(x,M\right);$ (2.8)
$\displaystyle\left|\partial^{\alpha}\left(\vec{P}_{1}-\vec{P}_{2}\right)\left(x\right)\right|\leq
M\delta^{m-\left|\alpha\right|}\text{ for }\left|\alpha\right|\leq m;$ (2.9)
$\displaystyle\left|\partial^{\alpha}Q_{i}(x)\right|\leq\delta^{-\left|\alpha\right|}\text{
for }\left|\alpha\right|\leq m,i=1,2;\text{ and}$ (2.10) $\displaystyle
Q_{1}\odot_{x}Q_{1}+Q_{2}\odot_{x}Q_{2}=1.$

Then

(2.11)
$\sum_{i=1,2}Q_{i}\odot_{x}Q_{i}\odot_{x}\vec{P}_{i}\in\Gamma(x,C_{w}M).$

It is clear that any $(C_{w},\delta_{\max})$-convex shape field is also
$(C_{w}^{\prime},\delta_{\max}^{\prime})$-convex for any $C_{w}^{\prime}\geq
C_{w}$ and $0<\delta_{\max}^{\prime}\leq\delta_{\max}$.

###  2.1. Solution to Problem 1.3

Using a standard argument involving partition of unity, we may assume that $E$
is compact.

Consider the bundle $(\Gamma_{\mathcal{K}}(x))_{x\in E}$ with fiber

(2.12) $\Gamma_{\mathcal{K}}(x)=\bigcup_{M\geq 0}\Gamma_{\mathcal{K}}(x,M),$

where

(2.13)
$\Gamma_{\mathcal{K}}(x,M)=\left\\{\vec{P}\in\vec{\mathcal{P}}:\,\,\begin{matrix}[l]\text{$\vec{P}(x)\in\mathcal{K}(x)$
and $\left|\partial^{\alpha}\vec{P}(x)\right|\leq M$ for
$\left|\alpha\right|\leq m$.}\end{matrix}\right\\}.$

A $C^{m}$-section of the bundle $(\Gamma_{\mathcal{K}}(x))_{x\in E}$ is
precisely a $C^{m}(\mathbb{R}^{n},\mathbb{R}^{d})$ function $\vec{F}$ such
that $\vec{F}(x)\in\mathcal{K}(x)$ for each $x\in E$.

To find a $C^{m}$-section of $(\Gamma_{\mathcal{K}}(x))_{x\in E}$, we use
Theorem 1.6. In particular, we will prove the following.

######  Lemma 2.1.

$\left(\Gamma_{\mathcal{K}}(x,M)\right)_{x\in E,M\geq 0}$ is a closed,
regular, $(C,1)$-convex shape field.

###### Proof.

We check each condition one at a time.

Shape field. It is clear that $\left(\Gamma_{\mathcal{K}}(x,M)\right)_{x\in
E,M\geq 0}$ is a closed shape field.

Regularity. Now we show that $\left(\Gamma_{\mathcal{K}}(x,M)\right)_{x\in
E,M\geq 0}$ is regular as in Definition 2.1. In particular, we want to show
that $\left(\Gamma_{\mathcal{K}}(x,M)\right)_{x\in E,M\geq 0}$ satisfies
conditions ((2.1))–((2.5)).

It is clear that $\left(\Gamma_{\mathcal{K}}(x,M)\right)_{x\in E,M\geq 0}$
satisfies condition ((2.1)).

Now we check condition ((2.3)). Let $\epsilon>0$ and $M\geq 0$ be given. Let
$\delta=\delta(\epsilon,M)$ be a small number to be chosen. Let
$x,x^{\prime}\in E$ with $\left|x-x^{\prime}\right|\leq\delta$,
$\vec{P}\in\Gamma_{\mathcal{K}}(x,M)$, and
$\vec{P}^{\prime}\in\Gamma_{\mathcal{K}}(x^{\prime})=\bigcup_{M\geq
0}\Gamma_{\mathcal{K}}(x^{\prime},M)$ satisfy

(2.14)
$\left|\partial^{\alpha}(\vec{P}-\vec{P}^{\prime})(x)\right|,\left|\partial^{\alpha}(\vec{P}-\vec{P}^{\prime})(x)\right|\leq\delta\left|x-x^{\prime}\right|^{m-\left|\alpha\right|}\text{
for }\left|\alpha\right|\leq m$

Since $\vec{P}^{\prime}\in\Gamma_{\mathcal{K}}(x^{\prime})$, we have

(2.15) $\vec{P}^{\prime}(x^{\prime})\in\mathcal{K}(x^{\prime}).$

By Taylor’s theorem and the fact that $\vec{P}\in\Gamma_{\mathcal{K}}(x,M)$ is
an $m$-jet, we have

(2.16)
$\left|\partial^{\alpha}\vec{P}(x^{\prime})\right|\leq\begin{cases}M(1+C\left|x-x^{\prime}\right|^{m-\left|\alpha\right|})\leq
M(1+C\delta^{m-\left|\alpha\right|})&\text{ for $\left|\alpha\right|<m$}\\\
M&\text{ for $\left|\alpha\right|=m$}\end{cases},$

where $C=C(m,n,d)$.

We see from (2.14) and (2.16) that

(2.17)
$\begin{split}\left|\partial^{\alpha}\vec{P}^{\prime}(x^{\prime})\right|&\leq\left|\partial^{\alpha}(\vec{P}-\vec{P^{\prime}})(x^{\prime})\right|+\left|\partial^{\alpha}\vec{P}(x^{\prime})\right|\\\
&\leq\begin{cases}M(1+C\delta^{m-\left|\alpha\right|})+\delta^{m+1-\left|\alpha\right|}&\text{
for $\left|\alpha\right|<m$}\\\ M+\delta&\text{ for
$\left|\alpha\right|=m$}\end{cases}.\end{split}$

In view of (2.15) and (2.17) while choosing $\delta$ to be sufficiently small
in a manner dependent only on $M$ and $C=C(m,n,d)$, we can conclude that
$\vec{P}^{\prime}\in\Gamma_{\mathcal{K}}(x^{\prime},M+\epsilon)$.

We now turn to condition ((2.5)).

Let $\epsilon>0$ and $M\geq 0$. Let $\delta\in(0,\epsilon)$. Let $x\in E$,
$\vec{P}\in\Gamma_{\mathcal{K}}(x,M)$,
$\vec{P}^{\prime}\in\Gamma_{\mathcal{K}}(x)=\bigcup_{M\geq
0}\Gamma_{\mathcal{K}}(x,M)$. Assume that
$\left|\partial^{\alpha}(\vec{P}-\vec{P}^{\prime})(x)\right|\leq\delta<\epsilon$.
We immediately see that
$\vec{P}^{\prime}\in\Gamma_{\mathcal{K}}(x,M+\epsilon)$. Condition ((2.5)) is
satisfied.

Therefore, $\left(\Gamma_{\mathcal{K}}(x,M)\right)_{x\in E,M\geq 0}$ is
regular.

$(C,1)$-convexity. Now we show that $(\Gamma_{\mathcal{K}}(x,M))_{x\in E,M\geq
0}$ is $(C,1)$-convex. Fix the following:

(2.18) $\displaystyle\delta\in(0,1],\,x\in E,\,M\geq 0;$ (2.19)
$\displaystyle\vec{P}_{1},\vec{P}_{2}\in\Gamma_{\mathcal{K}}(x,M)\text{ with
}\left|\partial^{\alpha}(\vec{P}_{1}-\vec{P}_{2})(x)\right|\leq
M\delta^{m-\left|\alpha\right|}\text{ for }\left|\alpha\right|\leq m;$ (2.20)
$\displaystyle Q_{1},Q_{2}\in\mathcal{P}\text{ with
}\left|\partial^{\alpha}Q_{i}(x)\right|\leq\delta^{-\left|\alpha\right|}\text{
and }Q_{1}\odot_{x}Q_{1}+Q_{2}\odot_{x}Q_{2}=1.$

We want to show that

(2.21)
$\vec{P}:=\sum_{i=1,2}Q_{i}\odot_{x}Q_{i}\odot_{x}\vec{P}_{i}\in\Gamma_{\mathcal{K}}(x,CM).$

Thanks to (2.20), we have $\vec{P}(x)\in\mathcal{K}(x)$. Thanks to (2.19) and
(2.20), we have $\left|\partial^{\alpha}\vec{P}(x)\right|\leq CM$. Therefore,
(2.21) holds.

∎

##  3\. Termination Lemma

Let $D=\dim\vec{\mathcal{P}}$. The purpose of this section is to prove the
following lemma, which serves as part 1 of Theorem 1.6.

######  Lemma 3.1 (Termination of Glaser Refinement).

If $(H(x))_{x\in E}$ is a convex bundle, then $H_{l}(x)=H_{2D+1}(x)$ for all
$l\geq 2D+1$.

The origin of Lemma 3.1 goes back to Glaeser [18]. Bierstone-Milman and
Fefferman also adapted it in their works [1] and [4].

Before beginning the proof of Lemma 3.1, we will show that the convexity
hypothesis on the holding spaces extends to the Glaeser refinements.

######  Lemma 3.2.

If $(H_{0}(x))_{x\in E}$ is a convex bundle, then $(H_{l}(x))_{x\in E}$ is a
convex bundle for all $l\geq 1$.

###### Proof.

We induct on $l\geq 0$. The case $l=0$ is trivial by assumption.

Let $l\geq 1$. Suppose $H_{l-1}(x)$ is convex for each $x\in E$. We want to
show that $H_{l}(x)$ is convex for each $x\in E$.

Fix $x_{0}\in E$. Let $\vec{P}_{0},\vec{Q}_{0}\in H_{l}(x_{0})$. Let
$\theta\in[0,1]$, and set

$\vec{R}_{0}:=(1-\theta)\vec{P}_{0}+\theta\vec{Q}_{0}.$

We want to show that $\vec{R}_{0}\in H_{l}(x_{0})$.

Let $\epsilon>0$. Since $\vec{P}_{0},\vec{Q}_{0}\in H_{l}(x_{0})$, there
exists $\delta>0$ such that, for any $x_{1},\cdots,x_{{k^{\sharp}}}\in E\cap
B(x_{0},\delta)$, there exist
$\vec{P}_{1},...,\vec{P}_{{k^{\sharp}}},\vec{Q}_{1},...,\vec{Q}_{k^{\sharp}}$,
with

(3.1) $\vec{P}_{j},\vec{Q}_{j}\in H_{l-1}(x_{j})\text{ for
}j=1,\cdots,{k^{\sharp}};$ (3.2)
$\left|\partial^{\alpha}(\vec{P}_{i}-\vec{P}_{j})(x_{j})\right|\leq\epsilon\left|x_{i}-x_{j}\right|^{m-\left|\alpha\right|}\text{
for }\left|\alpha\right|\leq m,0\leq i,j\leq{k^{\sharp}};$

and

(3.3)
$\left|\partial^{\alpha}(\vec{Q}_{i}-\vec{Q}_{j})(x_{j})\right|\leq\epsilon\left|x_{i}-x_{j}\right|^{m-\left|\alpha\right|}\text{for
}\left|\alpha\right|\leq m,0\leq i,j\leq{k^{\sharp}}.$

We set

(3.4) $\vec{R}_{j}:=(1-\theta)\vec{P}_{j}+\theta\vec{Q}_{j}\text{ for
}j=1,\cdots,{k^{\sharp}}.$

By the induction hypothesis that $H_{l-1}(x)$ is convex for each $x\in E$,
(3.1) and (3.4) imply $\vec{R}_{j}\in H_{l-1}(x_{j})$ for each
$j=1,\cdots,{k^{\sharp}}$. Moreover, by (3.4) and the triangle inequality,

(3.5)
$\displaystyle\left|\partial^{\alpha}(\vec{R}_{i}-\vec{R}_{j})(x_{j})\right|$
$\displaystyle\leq(1-\theta)\left|\partial^{\alpha}(\vec{P}_{i}-\vec{P}_{j})(x_{j})\right|+\theta\left|\partial^{\alpha}(\vec{Q}_{i}-\vec{Q}_{j})(x_{j})\right|$
(3.6)
$\displaystyle\leq\epsilon\left|x_{i}-x_{j}\right|^{m-\left|\alpha\right|}$

for $\left|\alpha\right|\leq m$ and $0\leq i,j\leq{k^{\sharp}}$.

Hence, $H_{l}(x_{0})$ is convex. Since $x_{0},\vec{P}_{0},$ and $\vec{Q}_{0}$
were arbitrary, the lemma is proved. ∎

In [1] and [6], the analogous version of Lemma 3.1 for affine holding spaces
was proven using an argument which relied on the well-definition of the
dimension of vector subspaces. Here, we adapt this argument to the case of
non-affine holding spaces so another definition of dimension will be required.

In this section, we will use the notation $B_{\eta}(\vec{P})$ to denote the
open ball of radius $\eta$ in $\vec{\mathcal{P}}$ centered $\vec{P}$ with
respect to the metric

(3.7) $d(\vec{P},\vec{P}^{\prime}):=\max_{|\alpha|\leq
m}|\partial^{\alpha}(\vec{P}-\vec{P}^{\prime})(0)|.$

$\overline{B}_{\eta}(\vec{P})$ will denote the analogous closed ball in
$\vec{\mathcal{P}}$.

Let $(H(x))_{x\in E}$ be a holding space with Glaeser refinement
$(\tilde{H}(x))_{x\in E}$. Observe that whenever $x_{0},x_{1},x_{2},...\in E$,
$\vec{P}_{0}\in\tilde{H}(x_{0})$ and $x_{j}\to x_{0}$ as $n\to\infty$, there
exist $\vec{P}_{j}\in H(x_{j})$ such that $d(\vec{P}_{j},\vec{P}_{0})\to 0$.
This property, rather than the definition of Glaeser refinement, will be key
in the proof of Lemma 3.1.

######  Definition 3.1.

Given $K\subset\vec{\mathcal{P}}$ convex and $\vec{P}\in\vec{\mathcal{P}}$, we
define the local dimension of $K$ at $\vec{P}$, denoted $\dim_{\vec{P}}K$, to
be the largest integer $k$ such that the following holds:

  * (3.8)

There exists $\eta>0$ such that $K\cap B_{\eta}(\vec{P})\supset W\cap
B_{\eta}(\vec{P})$ for some $k$-dimensional affine subspace $W\ni\vec{P}$.

We take $\dim_{\vec{P}}K=-\infty$ in the case where $\vec{P}\notin K$. Thus,
the condition $\dim_{\vec{P}}K\neq-\infty$ implies $\vec{P}\in K$.

For an example illustrating the notion of local dimension, consider the half-
ball $K=\\{(x,y)\in\mathbb{R}^{2}:x^{2}+y^{2}\leq 1,y\geq 0\\}$. In this case
$\dim_{(0,1/2)}K=2$ since $B_{1/10}((0,1/2))\subset K$ and $\dim_{(0,0)}K=1$
since any 2-dimensional ball containing $(0,0)$ must intersect the lower half-
plane but the line segment from $(-1/2,0)$ to $(1/2,0)$ is contained in $K$.
Furthermore, $\dim_{(1,0)}K=\dim_{(0,1)}K=0$ since in either case, any line
segment with the given point as its midpoint must intersect $K^{c}$.

######  Remark 3.2.

  1. (1)

If $\vec{P}$ is in the relative interior of $K$, written
$\vec{P}\in\text{int}K$, then we may take $K\cap B_{\eta}(\vec{P})=W\cap
B_{\eta}(\vec{P})$ in ((3.8)).

  2. (2)

In the case where $K$ is a closed, convex subset of $\vec{\mathcal{P}}$, then
it has a well-defined dimension as a set, often defined as the smallest
dimension of an affine subspace containing the set $K$, which we denote $\dim
K$. When $\vec{P}\in\text{int}K$, $\dim_{\vec{P}}K=\dim K$; in particular, for
a vector subspace $V\subset\vec{\mathcal{P}}$, $\dim_{\vec{P}}V=\dim V$ for
any $\vec{P}\in V$.

  3. (3)

Whenever $K\supset L$, we have $\dim_{\vec{P}}K\geq\dim_{\vec{P}}L$.

Lemma 3.1 will follow shortly from the following result:

######  Lemma 3.3.

Suppose $\left(H(x)\right)_{x\in E}$ is a convex bundle. Let $x\in E$,
$\vec{P}\in\vec{\mathcal{P}}$, and $k\geq 0$ be an integer. If

(3.10) $\dim_{\vec{P}}H_{2k+1}(x)\geq D-k,$

then $\dim_{\vec{P}}H_{l}(x)=\dim_{\vec{P}}H_{2k+1}(x)$ for all $l\geq 2k+1$.

To see the value of Lemma 3.3, let us use it to prove Lemma 3.1:

###### Proof of Lemma 3.1 via Lemma 3.3.

Let $x\in E$ and $\vec{P}\in H_{2D+1}(x)$. Then,
$\dim_{\vec{P}}H_{2D+1}(x)\geq 0$ because the zero-dimensional affine space
$W=\\{\vec{P}\\}$ satisfies $B_{\eta}(\vec{P})\cap K\supseteq
B_{\eta}(\vec{P})\cap W$ for any $\eta>0$.

Taking $k=D$ in (3.10), $\dim_{\vec{P}}H_{l}(x)=\dim_{\vec{P}}H_{2k+1}(x)\geq
0$ for all $l\geq 2k+1$. The local dimension is a nonnegative integer if and
only if $\vec{P}\in H_{l}(x)$; therefore, $\vec{P}\in H_{l}(x)$ for all $l\geq
2D+1\geq 2k+1$. ∎

Now, it suffices to prove Lemma 3.3.

###### Proof of Lemma 3.3.

We prove this result by induction on $k$. By Lemma 3.2, $H_{l}(y)$ is convex
for all $y\in E$ and $l\geq 0$. This convexity will be used throughout the
proof.

Base case ($k=0$): First, suppose $\dim_{\vec{P}}H_{1}(x)=D$ for some $x\in E$
and $\vec{P}\in H_{1}(x)$. In such a case $\dim_{\vec{P}}H(x)=D$ as well by
Remark 3.2 No. 3. Pick $\eta$ so $H_{1}(x)\cap B_{\eta}(\vec{P})\supset W\cap
B_{\eta}(\vec{P})$ for some $D$-dimensional affine subspace $W$ running
through $\vec{P}$. The only $D$-dimensional affine space in
$\vec{\mathcal{P}}$ is $\vec{\mathcal{P}}$ itself, so
$B_{\eta}(\vec{P})\subset H_{1}(x)$.

######  Claim 3.3.

$B_{\eta/2}(\vec{P})\subset H(y)$ for all $y$ sufficiently close to $x$.

###### Proof of Claim 3.3.

Suppose the contrary. Then, there is a sequence $x_{n}\to x$ and
$\vec{P}_{n}\in B_{\eta/2}(\vec{P})$ such that $\vec{P}_{n}\notin H(x_{n})$.

Consider the set $B_{\eta}(\vec{P})\cap H(x_{n})$. Since $\vec{P}_{n}\notin
H(x_{n})$ and $H(x_{n})$ is convex, there exists $\vec{Q}_{n}\in
B_{\eta}(\vec{P})$ such that $B_{\eta/4}(\vec{Q}_{n})$ and $H(x_{n})\cap
B_{\eta}(\vec{P})$ are disjoint subsets of $B_{\eta}(\vec{P})$. By passing to
a subsequence, we may assume there exists $\vec{Q}\in B_{\eta}(\vec{P})$ such
that $d(\vec{Q}_{n},\vec{Q})\to 0$. This contradicts the fact that $\vec{Q}\in
H_{1}(x)$, thus proving Claim 3.3. ∎

Since $B_{\eta/2}(\vec{P})\subset H(y)$ for all $y$ sufficiently close to $x$,
we see this property also holds for $H_{1}(y)$ in place of $H(y)$. In fact,
let $r>0$; then for $y$ in an open subset of $E$, $H_{l+1}(y)\cap
B_{r}(\vec{P})$ is determined solely by $H_{l}(z)\cap B_{r}(\vec{P})$ for $z$
in that open subset by the definition of Glaeser refinement.

Repeating this logic, we see $B_{\eta/2}(\vec{P})\subset H_{l}(y)$ for all
$H_{l}(y)$ with $l\geq 1$ and $y\in E$ sufficiently close to $x$; this
includes the case $y=x$. Thus, the local dimension of $H_{l}(x)$ at $\vec{P}$
remains at $D$ for all $l$, establishing the base case.

Induction Hypothesis: Assume

$\dim_{\vec{P}}H_{2j+1}(x)\geq D-j$

implies $\dim_{\vec{P}}H_{l}(x)=\dim_{\vec{P}}H_{2j+1}(x)$ for all $l\geq
2j+1$ whenever $j\leq k$.

Now suppose $\dim_{\vec{P}}H_{2k+3}(x)\geq D-k-1$. If
$\dim_{\vec{P}}H_{2k+3}(x)>D-k-1,$ then the conclusion follows from the
induction hypothesis. Thus, we may assume

(3.11) $\dim_{\vec{P}}H_{2k+3}(x)=D-k-1.$

Furthermore,

(3.12)
$\dim_{\vec{P}}H_{2k+1}(x)=\dim_{\vec{P}}H_{2k+2}(x)=\dim_{\vec{P}}H_{2k+3}(x)=D-k-1,$

since if $\dim_{\vec{P}}H_{2k+1}(x)>D-k-1$, $\dim_{\vec{P}}H_{l}(x)$ would
have terminated at some number greater than $D-k-1$ starting at some $l<2k+1$
by induction hypothesis. We want to show

(3.13) $\dim_{\vec{P}}H_{l}(x)=D-k-1$

whenever $l\geq 2k+3$.

######  Claim 3.4.

Let $l=2k+1,2k+2$. Fix $x\in E$ and $\vec{P}\in H_{l+1}(x)$. Pick $\eta>0$ so
that $H_{l+1}(x)\cap B_{\eta}(P)\supseteq W\cap B_{\eta}(P)$ for some
$(D-k-1)$-dimensional affine subspace $W\ni P$. For $y$ sufficiently close to
$x$, either

(3.14) $\dim[H_{l}(y)\cap B_{\eta}(\vec{P})]>D-k-1$

or

(3.15) $H_{l}(y)\cap B_{\eta/2}(\vec{P})=V_{y}\cap B_{\eta/2}(\vec{P})$

for some $(D-k-1)$-dimensional affine subspace $V_{y}$ of $\vec{\mathcal{P}}$.
Furthermore, if $\vec{P}$ is in the relative boundary of $H_{l+1}(x)$, then in
particular, (3.14) holds for $y$ sufficiently close to $x$.

###### Proof of Claim 3.4.

Since $l=2k+1,2k+2$, we have
$\dim_{\vec{P}}H_{l}(x)=\dim_{\vec{P}}H_{l+1}(x)=D-k-1$.

If $\vec{P}\in\text{int}H_{l+1}(x)$, then suppose the contrary. That is, there
exists a sequence $x_{n}\to x$ such that (3.14) and (3.15) fail for $y=x_{n}$.
In particular, since (3.14) does not hold, $H_{l}(x_{n})\cap B_{\eta}(P)$ is
contained in a $(D-k-1)$-dimensional affine space $V_{n}$. Here, we use Remark
3.2 No. 1.

By compactness of the Grassmannian of $(D-k-1)$-planes in $\vec{\mathcal{P}}$
with intersection in $\overline{B}_{\eta}(\vec{P})$ and the closure of
$\overline{B}_{\eta}(\vec{P})$, we may assume $V_{n}$ converges to a subspace
$V$ by passing to a subsequence.

Since $H_{l+1}(x)\cap B_{\eta}(\vec{P})=W\cap B_{\eta}(\vec{P})$, $V$ must
equal $W$. Else, there exists $\vec{P}^{\prime}\in H_{l+1}(x)$ which is not
the limit of $\vec{P}_{n}\in H_{l}(x_{n})$, contradicting our prior
observation on refinements.

Since (3.15) fails, there exists $\vec{P}^{\prime}_{n}\in V_{n}\cap
B_{\eta/2}(\vec{P})$ such that $\vec{P}^{\prime}_{n}\notin H_{l}(x_{n})$.
Furthermore, by the convexity of $H_{l}(x_{n})$, there exists $\vec{Q}_{n}\in
B_{\eta}(\vec{P})\cap V_{n}$ such that $\vec{Q}_{n}\notin H_{l}(x_{n})$ and
$B_{\eta/4}(\vec{Q}_{n})\subset B_{\eta}(\vec{P})$ is disjoint from
$H_{l}(x_{n})$.

By passing to another subsequence, we may assume $\vec{Q}_{n}$ converges to
some $\vec{Q}\in V=W$. However, this contradicts the fact that $\vec{Q}\in
H_{l+1}(x)$, establishing our claim for $\vec{P}\in\text{int}H_{l+1}(x)$.

If $\vec{P}$ is in the relative boundary of $H_{l+1}(x)$, then $\dim
H_{l+1}(x)>D-k-1$. Taking $\vec{Q}\in\text{int}H_{l+1}(x)$ such that
$\dim_{\vec{Q}}H_{l+1}(x)>D-k-1$, we may repeat the above argument to deduce,
at the very least, that for $y$ sufficiently close to $x$, $\dim[H_{l}(y)\cap
B_{\eta}(\vec{P})]>D-k-1$. This completes the proof of Claim 3.4. ∎

Consider $y$ sufficiently close to $x$. There are now distinct cases based on
which of (3.14) or (3.15) holds for $l=2k+1$ and $l=2k+2$.

First, there is the case that (3.14) holds for $l=2k+1$ or $l=2k+2$, which
means $H_{j}(y)\cap B_{\eta}(\vec{P})$ and $H_{l}(y)\cap B_{\eta}(\vec{P})$
have equal relative interiors for all $j\geq l$ by induction hypothesis.

Else, (3.15) holds for both $l=2k+1,2k+2$, meaning $H_{2k+1}(y)\cap
B_{\eta/2}(\vec{P})=V_{y}\cap B_{\eta/2}(\vec{P})$ for some
$(D-k-1)$-dimensional affine subspace $V_{y}$ of $\vec{\mathcal{P}}$ and
$H_{2k+2}(y)\cap B_{\eta/2}(\vec{P})=V_{y}^{\prime}\cap B_{\eta/2}(\vec{P})$
for some $(D-k-1)$-dimensional affine subspace $V_{y}^{\prime}$ of
$\vec{\mathcal{P}}$. We see that $V_{y}=V_{y}^{\prime}$ since
$H_{2k+2}(y)\subset H_{2k+1}(y)$ and $\dim V_{y}=\dim V_{y}^{\prime}$.

In any case, we see that for $y$ sufficiently close to $x$, $H_{2k+1}(y)\cap
B_{\eta}(\vec{P})$ and $H_{2k+2}(y)\cap B_{\eta}(\vec{P})$ have equal relative
interiors. Ideally, one would like to show that these sets are completely
equal; however, Lemma 3.4 below will address this disparity.

######  Lemma 3.4.

Let $(H(x))_{x\in E}$ be a bundle and fix $y_{0}\in E$. Fix $\delta_{0}>0$.
Let $\left(\hat{H}(x)\right)_{x\in E}$ denote the bundle made from taking
$(H(x))_{x\in E}$ and replacing $H(y)$ by $\overline{H}(y)$ for all $y\in
B(y_{0},\delta_{0})\setminus\\{y_{0}\\}$. Then
$H_{1}(y_{0})=\hat{H}_{1}(y_{0})$.

In particular, if two bundles have the same relative interiors of their fibers
at every $y\in B(y_{0},\delta_{0})\setminus\\{y_{0}\\}$ and $\vec{P}$ is in
both their fibers at $y_{0}$, then $\vec{P}$ is either in the fiber at $y_{0}$
of both Glaeser refinements or neither.

We recall that given $r>0$, $y$ in a relatively open subset of $E$,
$H_{l+1}(y)\cap B_{r}(\vec{P})$ is determined solely by $H_{l}(z)\cap
B_{r}(\vec{P})$ for other $z$ in that relatively open subset. Applying Lemma
3.4 with $y_{0}$ ranging over $y$ sufficiently close to $x$, we find

(3.16) $\text{int}H_{l}(y)=\text{int}H_{2k+1}(y)\text{ for }l\geq 2k+1.$

Another application of Lemma 3.4 shows that $H_{l}(x)\cap B_{\eta/2}(\vec{P})$
remains constant in $l$ for $l\geq 2k+1$ as well, completing the proof of
Lemma 3.3. ∎

At this point, it merely remains to prove Lemma 3.4.

###### Proof of Lemma 3.4.

Since $H(x)\subseteq\hat{H}(x)$ for all $x\in E$, we have
$H_{1}(y_{0})\subseteq\hat{H}_{1}(y_{0})$.

Suppose $\vec{P}_{0}\in\hat{H}_{1}(y_{0})$. Let $\epsilon>0$ and pick
$0<\delta<\delta_{0}$ such that for every $y_{1},...,y_{k^{\sharp}}\in
B(y_{0},\delta)$, there exists $\vec{P}_{j}\in\hat{H}(y_{j})$ for
$j=0,,1,...,k^{\sharp}$ such that

(3.17)
$|\partial^{\alpha}(\vec{P}_{i}-\vec{P}_{j})(y_{j})|\leq(\epsilon/2)|y_{i}-y_{j}|^{m-|\alpha|}\text{
for }|\alpha|\leq m,0\leq i,j\leq k^{\sharp}.$

Now fix a particular choice of $y_{1},...,y_{k^{\sharp}}$ and make a choice of
$\vec{P}_{j}\in\hat{H}(y_{j})$ satisfying (3.17). If all the $\vec{P}_{j}$ lie
in $H(y_{j})$, then we are done. It is possible to choose $\vec{P}_{j}\notin
H(y_{j})$; however, we may always choose $\vec{P}_{j}^{\prime}\in H(y_{j})$
such that
$|\partial^{\alpha}(\vec{P}_{j}-\vec{P}_{j}^{\prime})(x)|\leq\eta_{j}$ for
some $\eta_{j}>0$ and all $x\in E,0\leq|\alpha|\leq m$.

By the triangle inequality, for any $|\alpha|\leq m,0\leq
i,j\leq{k^{\sharp}}$,

(3.18)
$\displaystyle|\partial^{\alpha}(\vec{P}_{i}^{\prime}-\vec{P}_{j}^{\prime})(y_{j})|$
$\displaystyle\leq|\partial^{\alpha}(\vec{P}_{i}^{\prime}-\vec{P}_{i})(y_{j})|+|\partial^{\alpha}(\vec{P}_{i}-\vec{P}_{j})(y_{j})|+|\partial^{\alpha}(\vec{P}_{j}-\vec{P}_{j}^{\prime})(y_{j})|$
(3.19)
$\displaystyle\leq\eta_{i}+(\epsilon/2)|y_{i}-y_{j}|^{m-|\alpha|}+\eta_{j}$
(3.20) $\displaystyle\leq\epsilon|y_{i}-y_{j}|^{m-|\alpha|}$

by taking $\eta_{j}$ ($0\leq j\leq{k^{\sharp}}$) sufficiently small relative
to $\epsilon$ and $\min_{i\neq j}|y_{i}-y_{j}|$.

Since the choices of $\epsilon$ and $y_{j}$ were arbitrary, this shows
$\vec{P}_{0}\in H_{1}(y_{0})$.

The second conclusion follows from the fact that if two fibers have the same
relative interior, then they have the same closure and by considering the
holding space $(H^{\prime}(x))_{x\in E}$, where $H^{\prime}(x)=H(x)$ for
$x\neq y_{0}$ and $H(y_{0})=\\{\vec{P}\\}$. ∎

##  4\. Reduction to the scalar-valued case

In this section, we show that the validity of Theorem 1.6 for
$C^{m+d}(\mathbb{R}^{n},\mathbb{R})$ implies the same for
$C^{m}(\mathbb{R}^{n},\mathbb{R}^{d})$, thus reducing all future analysis to
scalar-valued functions. The main lemma of the section is the following.

######  Lemma 4.1.

Let $m,n,d$ be positive integers. Let ${k^{\sharp}}$ be a sufficiently large
constant depending only on $m,n,d$. If Theorem 1.6 holds for
$C^{m+1}(\mathbb{R}^{n+d},\mathbb{R})$ and shape fields of type $(m+1,n+d,1)$,
then it also holds for $C^{m}(\mathbb{R}^{n},\mathbb{R}^{d})$ and shape fields
of type $(m,n,d)$.

We use a gradient trick inspired by [15].

To begin with, we put more emphasis on the degree of regularity and dimension:
For positive integers $s_{1},s_{2},s_{3}$, we write
$\mathcal{P}^{s_{1}}(\mathbb{R}^{s_{2}},\mathbb{R}^{s_{3}})$ to denote the
vector space of $s_{3}$-tuples of polynomials with degree no greater than
$s_{1}$ on $\mathbb{R}^{s_{2}}$. We write $\odot_{x}^{s_{1},s_{2}}$ to denote
the ring multiplication of $s_{1}$-jets at $x\in\mathbb{R}^{s_{2}}$, and
denote this ring by $\mathcal{R}_{x}^{s_{1},s_{2}}$. As before, we use the
same notation to denote the action of $\mathcal{R}_{x}^{s_{1},s_{2}}$ on the
module $\mathcal{P}^{s_{1}}(\mathbb{R}^{s_{2}},\mathbb{R}^{s_{3}})$.

For the rest of the section, we use
$(x,v)=(x_{1},\cdots,x_{n},v_{1},\cdots,v_{d})$ to denote a vector in
$\mathbb{R}^{n+d}$. We write $\nabla_{v}$ to denote the operator
$(\partial_{v_{1}},\cdots,\partial_{v_{d}})$. Note that if $P$ is an
$(m+1)$-jet (of a real-valued function) on $\mathbb{R}^{n+d}$, then
$P(\,\cdot\,,0)$ is an $(m+1)$-jet (of a real-valued function) on
$\mathbb{R}^{n}$, and $\nabla_{v}\big{|}_{v=0}P$ is an $m$-jet (of an
$\mathbb{R}^{d}$-valued function) on $\mathbb{R}^{n}$.

We say a shape field $(\Gamma(x,M))_{x\in E,M\geq 0}$ is of type
$(s_{1},s_{2},s_{3})$ if
$\Gamma(x,M)\subset\mathcal{P}^{s_{1}}(\mathbb{R}^{s_{2}},\mathbb{R}^{s_{3}})$
for all $x\in E$ and $M\geq 0$.

######  Lemma 4.2.

Let $E\subset\mathbb{R}^{n}$ be a compact set. Let $(\Gamma(x,M))_{x\in
E,M\geq 0}$ be a regular $(C_{w},\delta_{\text{max}})$-convex shape field of
type $(m,n,d)$. For each $x\in E$ and $M\geq 0$, define

(4.1)
$\widehat{\Gamma}((x,0),M)=\left\\{\widehat{P}\in\mathcal{P}^{m+1}(\mathbb{R}^{n+d},\mathbb{R}):\,\begin{matrix}[l]\left|\partial^{\alpha}\widehat{P}(x,0)\right|\leq
M\text{ for }\alpha\in\mathbb{N}_{0}^{n+d},\left|\alpha\right|\leq m+1,\\\
\widehat{P}(\,\cdot\,,0)\equiv 0\text{, and
}\nabla_{v}\big{|}_{v=0}\widehat{P}\in\Gamma(x,M)\end{matrix}\right\\}.$

There exists $C=C(m,n,d,C_{w})$ such that

  1. (1)

$(\widehat{\Gamma}((x,0),M))_{(x,0)\in E\times\left\\{0\right\\},M\geq 0}$ is
a regular $(C,\delta_{\text{max}})$-convex shape field of type $(m+1,n+d,1)$
on $E\times\left\\{0\right\\}\subset\mathbb{R}^{n+d}$.

  2. (2)

$C^{-1}\|\Gamma\|\leq\|\widehat{\Gamma}\|\leq C\|\Gamma\|$, with $\|\cdot\|$
as in Definition 1.5.

As a corollary to part 2 of the above lemma, the Finiteness Lemma (Lemma 6.1)
for vector-valued functions immediately follows from the lemma for scalar-
valued functions.

###### Proof.

We begin with the first statement. It is clear that
$(\widehat{\Gamma}((x,0),M))_{(x,0)\in E\times\left\\{0\right\\},M\geq 0}$ is
a shape field of type $(m+1,n+d,1)$.

We now prove that $(\widehat{\Gamma}((x,0),M))_{(x,0)\in
E\times\left\\{0\right\\},M\geq 0}$ is regular.

Condition ((2.1)) is clearly satisfied, thanks to the first constraint in
(4.1).

We prove condition ((2.3)). Let $\epsilon>0$ and $M\geq 0$. Let
$\eta=\eta(\epsilon,M)>0$ be a small number to be chosen. Let
$(x,0),(x^{\prime},0)\in E\times\left\\{0\right\\}$ with
$\left|(x,0)-(x^{\prime},0)\right|\leq\eta$. Let
$\widehat{P}\in\widehat{\Gamma}((x,0),M)$ and
$\widehat{P}\in\widehat{\Gamma}(x^{\prime},0)=\bigcup_{M^{\prime}\geq
0}\widehat{\Gamma}((x^{\prime},0),M^{\prime})$, with

(4.2)
$\left|\partial^{\alpha}(\widehat{P}-\widehat{P}^{\prime})(x,0)\right|\,,\,\left|\partial^{\alpha}(\widehat{P}-\widehat{P}^{\prime})(x^{\prime},0)\right|\leq\delta\left|x-x^{\prime}\right|^{m+1-\left|\alpha\right|}\text{
for }\alpha\in\mathbb{N}_{0}^{n+d},\left|\alpha\right|\leq m+1.$

Set $\vec{P}=\nabla_{v}\big{|}_{v=0}\widehat{P}$ and
$\vec{P}^{\prime}=\nabla_{v}\big{|}_{v=0}\widehat{P}^{\prime}$. By the
definition of $\widehat{\Gamma}$, $\vec{P}\in\Gamma(x,M)$ and
$\vec{P}^{\prime}\in\Gamma(x^{\prime})$. In view of (4.2), we have

(4.3)
$\left|\partial^{\beta}(\vec{P}-\vec{P}^{\prime})(x)\right|\,,\,\left|\partial^{\beta}(\vec{P}-\vec{P}^{\prime})(x^{\prime})\right|\leq
C\eta\left|x-x^{\prime}\right|^{m-\left|\alpha\right|}\text{ for
}\alpha\in\mathbb{N}_{0}^{n},\left|\alpha\right|\leq m.$

Since $(\Gamma(x,M))_{x\in E,M\geq 0}$ is regular, for sufficiently small
$\eta$, (4.3) implies

(4.4) $\vec{P}^{\prime}\in\Gamma(x,M+\tilde{\epsilon}(\eta))\text{,
i.e.,}\nabla_{v}\big{|}_{v=0}\widehat{P}\in\Gamma(x,M+\tilde{\epsilon}(\eta)).$

Here $\tilde{\epsilon}(\cdot)$ is a decreasing function of $\eta$, depending
only on $M$.

We choose $\eta$ to be sufficiently small (depending only on $\epsilon$ and
$M$), so that (4.2), (4.4), and a similar argument as in Lemma 2.1 altogether
imply $\widehat{P}^{\prime}\in\widehat{\Gamma}((x^{\prime},0),M+\epsilon)$.
Condition ((2.3)) is satisfied.

For condition ((2.5)), let $\epsilon>0$, $M\geq 0$, and let $\eta>0$ be a
sufficiently small number to be determined. Let $(x,0)\in
E\times\left\\{0\right\\}$, $\widehat{P}\in\widehat{\Gamma}((x,0),M)$,
$\widehat{P}^{\prime}\in\widehat{\Gamma}(x,0)$ with

(4.5)
$\left|\partial^{\alpha}(\widehat{P}-\widehat{P}^{\prime})(x,0)\right|\leq\eta\text{
for }\alpha\in\mathbb{N}_{0}^{n+d},\,\left|\alpha\right|\leq m+1.$

Set $\vec{P}=\nabla_{v}\big{|}_{v=0}\widehat{P}$ and
$\vec{P}^{\prime}=\nabla_{v}\big{|}_{v=0}\widehat{P}^{\prime}$. By the
definition of $\widehat{\Gamma}$, $\vec{P}\in\Gamma(x,M)$ and
$\vec{P}^{\prime}\in\Gamma(x^{\prime})$. In view of (4.5), we have

(4.6) $\left|\partial^{\beta}(\vec{P}-\vec{P}^{\prime})(x)\right|\leq
C\eta\text{ for }\beta\in\mathbb{N}_{0}^{n},\,\left|\beta\right|\leq m.$

Since $(\Gamma(x,M))_{x\in E,M\geq 0}$ is regular, for sufficiently small
$\eta$ (depending only on $\epsilon$ and $M$), (4.5), (4.6), and a similar
argument as in Lemma 2.1 imply
$\widehat{P}^{\prime}\in\widehat{\Gamma}((x,0),M+\epsilon)$. Condition ((2.5))
is satisfied.

Thus, we have shown that $(\widehat{\Gamma}((x,0),M))_{(x,0)\in
E\times\left\\{0\right\\},M\geq 0}$ is regular.

We now show that $(\widehat{\Gamma}((x,0),M))_{(x,0)\in
E\times\left\\{0\right\\},M\geq 0}$ is $(C,\delta_{\text{max}})$-convex for
some $C$ depending only on $m,n,d,C_{w}$.

Let $\delta\in(0,\delta_{\text{max}}]$, $(x,0)\in E\times\left\\{0\right\\}$,
$M\geq 0$,
$\widehat{P}_{1},\widehat{P}_{2},\widehat{Q}_{1},\widehat{Q}_{2}\in\mathcal{P}^{m+1}(\mathbb{R}^{n+d},\mathbb{R})$
satisfy

(4.7)
$\displaystyle\widehat{P}_{1},\widehat{P}_{2}\in\widehat{\Gamma}((x,0),M)$
(4.8)
$\displaystyle\left|\partial^{\alpha}(\widehat{P}_{1}-\widehat{P}_{2})(x,0)\right|\leq
M\delta^{m+1-\left|\alpha\right|}\text{ for
}\alpha\in\mathbb{N}_{0}^{n+d},\,\left|\alpha\right|\leq m+1;$ (4.9)
$\displaystyle\left|\partial^{\alpha}\widehat{Q}_{i}(x,0)\right|\leq\delta^{-\left|\alpha\right|}\text{
for }\alpha\in\mathbb{N}_{0}^{n+d},\,\left|\alpha\right|\leq
m+1,\,i=1,2;\text{ and}$ (4.10)
$\displaystyle\widehat{Q}_{1}\odot_{(x,0)}^{m+1,n+d}\widehat{Q}_{1}+\widehat{Q}_{2}\odot_{(x,0)}^{m+1,n+d}\widehat{Q}_{2}=1.$

We want to show that

(4.11)
$\widehat{P}=\sum_{i=1,2}\widehat{Q}_{i}\odot_{(x,0)}^{m+1,n+d}\widehat{Q}_{i}\odot_{(x,0)}^{m+1,n+d}\widehat{P}_{i}\in\widehat{\Gamma}((x,0),CM),$

or equivalently,

(4.12) $\displaystyle\left|\partial^{\alpha}\widehat{P}(x,0)\right|\leq
M\text{ for }\alpha\in\mathbb{N}_{0}^{n+d},\,\left|\alpha\right|\leq m+1,$
(4.13) $\displaystyle\widehat{P}(\,\cdot\,,0)\equiv 0\text{, and }$ (4.14)
$\displaystyle\nabla_{v}\big{|}_{v=0}\widehat{P}\in\Gamma(x,CM).$

Thanks to (4.7), we have

(4.15) $\displaystyle\left|\partial^{\alpha}\widehat{P}_{i}(x,0)\right|\leq
M\text{ for }\alpha\in\mathbb{N}_{0}^{n+d},\,\left|\alpha\right|\leq m+1,$
(4.16) $\displaystyle\widehat{P}_{i}(\,\cdot\,,0)\equiv 0\text{ for
}i=1,2,\text{ and }$ (4.17)
$\displaystyle\nabla_{v}\big{|}_{v=0}\widehat{P}_{i}\in\Gamma(x,M)\text{ for
}i=1,2..$

In view of (4.10) and the definition of $\widehat{P}$ in (4.11), we have

(4.18)
$\partial^{\alpha}\widehat{P}(x,0)=\sum_{\beta+\gamma=\alpha}C_{\alpha,\beta,\gamma}\cdot\partial^{\beta}\widehat{Q}_{1}(x,0)\cdot\partial^{\gamma}\widehat{Q}_{1}(x,0)\cdot\partial^{\alpha-\beta-\gamma}(\widehat{P}_{1}-\widehat{P}_{2})(x,0).$

Using (4.8) and (4.9) to estimate (4.17), we see that (4.12) follows.

Thanks to (4.15), we have

$\widehat{P}(\,\cdot\,,0)=\sum_{i=1,2}\widehat{Q}_{i}\big{|}_{v=0}\odot_{(x,0)}^{m+1,n+d}\widehat{Q}_{i}\big{|}_{v=0}\odot_{(x,0)}^{m+1,n+d}\widehat{P}_{i}\big{|}_{v=0}\equiv
0.$

Thus, (4.13) follows.

Let $\pi:\mathcal{R}_{(x,0)}^{m+1,n+d}\to\mathcal{R}_{(x,0)}^{m,n+d}$ be the
natural projection. Thanks to (4.16), we have

(4.19)
$\begin{split}\nabla_{v}\big{|}_{v=0}\widehat{P}&=\sum_{i=1,2}\pi\widehat{Q}_{i}\big{|}_{v=0}\odot_{(x,0)}^{m,n+d}\pi\widehat{P}_{i}\big{|}_{v=0}\odot_{(x,0)}^{m,n+d}\nabla_{v}\big{|}_{v=0}\widehat{P}_{i}\\\
&\quad\quad\quad+2\pi\widehat{Q}_{i}\big{|}_{v=0}\odot_{(x,0)}^{m,n+d}\pi\widehat{P}_{i}\big{|}_{v=0}\odot_{(x,0)}^{m,n+d}\nabla_{v}\big{|}_{v=0}\widehat{Q}_{i}\\\
&=\sum_{i=1,2}\pi\widehat{Q}_{i}\big{|}_{v=0}\odot_{(x,0)}^{m,n+d}\pi\widehat{Q}_{i}\big{|}_{v=0}\odot_{(x,0)}^{m,n+d}\nabla_{v}\big{|}_{v=0}\widehat{P}.\end{split}$

We write $\partial^{\alpha}=\partial_{x}^{\beta}\partial_{v}^{\xi}$, for
$\alpha\in\mathbb{N}_{0}^{n+d}$, $\beta\in\mathbb{N}_{0}^{n}$,
$\xi\in\mathbb{N}_{0}^{d}$.

If $\left|\xi\right|=1$, i.e., $\partial_{v}=\partial_{v_{s}}$ for some
$s\in\left\\{1,\cdots,d\right\\}$, then (4.8) implies

$\left|\partial^{\alpha}(\widehat{P}_{1}-\widehat{P}_{2})(x,0)\right|=\left|\partial_{x}^{\beta}(\partial_{v_{s}}\big{|}_{v=0}\widehat{P}_{1}-\partial_{v_{s}}\big{|}_{v=0}\widehat{P}_{2})(x)\right|\leq
M\delta^{m-\left|\beta\right|}\text{ for }\left|\beta\right|\leq m.$

Therefore,

(4.20)
$\left|\partial_{x}^{\beta}\left[\nabla_{v}\big{|}_{v=0}(\widehat{P}_{1}-\widehat{P}_{2})\right](x)\right|\leq
CM\delta^{m-\left|\beta\right|}\text{ for }\left|\beta\right|\leq m.$

On the other hand, we see from (4.9) that

(4.21)
$\left|\partial_{x}^{\beta}\left[\pi\widehat{Q}_{i}\big{|}_{v=0}\right]\right|=\left|\left|\partial_{x}^{\beta}\widehat{Q}(x,0)\right|\leq\delta^{-\left|\beta\right|}\text{
for }\left|\beta\right|\leq m.\right|$

In view of (4.19)–(4.21) and the $(C_{w},\delta_{\text{max}})$-convexity of
$(\Gamma(x,M))_{x\in E,M\geq 0}$, we can conclude that
$\widehat{P}\in\widehat{\Gamma}((x,0),CM)$, so (4.11) holds. We have shown
that $(\widehat{\Gamma}((x,0),M))_{(x,0)\in E\times\left\\{0\right\\},M\geq
0}$ is $(C,\delta_{\text{max}})$-convex.

This concludes the proof of Lemma 4.2(1).

We turn to the second statement.

Let $M=2\|\widehat{\Gamma}\|$. We will show that $\|\Gamma\|\leq CM$.

Let $x_{1},\cdots,x_{k^{\sharp}}\in E$ be given. There exist
$\widehat{P}_{1}\in\widehat{\Gamma}((x_{1},0),M),\cdots,\widehat{P}_{k^{\sharp}}\in\widehat{\Gamma}((x_{k^{\sharp}},0),M)$
such that

(4.22)
$\left|\partial^{\alpha}(\widehat{P}_{i}-\widehat{P}_{j})(x_{i},0)\right|\leq
M\left|(x_{i},0)-(x_{j},0)\right|^{m+1-\left|\alpha\right|}\text{ for
$\alpha\in\mathbb{N}_{0}^{n+d}$, $\left|\alpha\right|\leq m+1$}.$

We set

$\vec{P}_{i}:=\nabla_{v}\big{|}_{v=0}\widehat{P}_{i}\in\mathcal{P}^{m}(\mathbb{R}^{n},\mathbb{R}^{d})\text{
for }1\leq i\leq{k^{\sharp}}.$

It follows immediately from (4.22) that

$\left|\partial^{\beta}(\vec{P}_{i}-\vec{P}_{j})(x_{i})\right|\leq
CM\left|x_{i}-x_{j}\right|^{m-\left|\beta\right|}\text{ for
$\beta\in\mathbb{N}_{0}^{n}$, $\left|\beta\right|\leq m$}.$

Therefore, $\|\Gamma\|\leq\|\widehat{\Gamma}\|$.

Now we show the reverse direction. Let $M=2\|\Gamma\|$. We will show that
$\|\widehat{\Gamma}\|\leq M$.

Let $x_{1},\cdots,x_{k^{\sharp}}\in E$ be given. There exist
$\vec{P}_{1}\in\Gamma(x_{1},M),\cdots,\vec{P}_{k^{\sharp}}\in\Gamma(x_{k^{\sharp}},M)$
such that

(4.23) $\left|\partial^{\beta}(\vec{P}_{i}-\vec{P}_{j})(x_{i})\right|\leq
M\left|x_{i}-x_{j}\right|^{m-\left|\alpha\right|}\text{ for
$\beta\in\mathbb{N}_{0}^{n}$, $\left|\beta\right|\leq m$}.$

We define

$\widehat{P}_{i}(\,\cdot\,,v)=v\cdot\vec{P}_{i}(\,\cdot\,)\in\mathcal{P}^{m+1}(\mathbb{R}^{n+d},\mathbb{R})\text{
for }1\leq i\leq{k^{\sharp}}.$

Observe that for $1\leq i\leq{k^{\sharp}}$, $\xi\in\mathbb{N}_{0}^{d}$,

(4.24) $\partial_{v}^{\xi}\big{|}_{v=0}\widehat{P}_{i}=\begin{cases}0&\text{
if }\left|\xi\right|=0\\\ P_{i,j}\text{ for some $1\leq j\leq d$}&\text{ if
}\left|\xi\right|=1\\\ 0&\text{ if }\left|\xi\right|\geq 2\end{cases}\,.$

Therefore, for $\left|\xi\right|\neq 1$, we have

(4.25)
$\left|\partial_{x}^{\beta}\partial_{v}^{\xi}(\widehat{P}_{i}-\widehat{P}_{j})(x_{i},0)\right|=0\text{
for
}\beta\in\mathbb{N}_{0}^{n},\,\xi\in\mathbb{N}_{0}^{d},\,\left|\beta\right|+\left|\xi\right|\leq
m+1.$

Suppose $\left|\xi\right|=1$. Without loss of generality, we may assume
$\partial_{v}^{\xi}=\partial_{v_{s}}$ for some
$s\in\left\\{1,\cdots,d\right\\}$. Thanks to (4.23), we have

(4.26)
$\begin{split}\left|\partial_{x}^{\beta}\partial_{v_{s}}(\widehat{P}_{i}-\widehat{P}_{j})(x_{i},0)\right|=\left|\partial_{x}^{\beta}(P_{i,s}-P_{j,s})(x_{i})\right|\leq
M\left|x_{i}-x_{j}\right|^{m-\left|\alpha\right|}=M\left|x_{i}-x_{j}\right|^{m+1-\left|\gamma\right|}.\end{split}$

We see from (4.25) and (4.26) that $\|\widehat{\Gamma}\|\leq M$. This proves
the second statement.

∎

With Lemma 4.2 in hand, we are ready to prove Lemma 4.1.

###### Proof of Lemma 4.1.

Let $Q_{0}\subset\mathbb{R}^{n}$ be a cube of length $3$ and $E\subset Q_{0}$
be compact. Let $\widehat{E}=E\times\left\\{0\right\\}\subset\mathbb{R}^{n+d}$
and $\widehat{Q}_{0}=Q_{0}\times[0,0+3\delta_{Q_{0}})^{d}$. Note that
$\widehat{Q}_{0}$ is a hypercube of sidelength less than 3.

Let $(\Gamma(x,M))_{x\in E,M\geq 0}$ be a regular $(C_{w},1)$-convex shape
field. Write $\Gamma(x)=\bigcup_{M\geq 0}\Gamma(x,M)$. Suppose
$\Gamma^{*}(x)\neq\varnothing$ for all $x\in E$, where $(\Gamma^{*}(x))_{x\in
E}$ is the termination of iterated Glaeser refinements of $(\Gamma(x))_{x\in
E}$ guaranteed by Lemma 3.3. We write
$\Gamma^{*}(x,M)=\Gamma^{*}(x)\cap\Gamma(x,M)$ for each $x\in E$ and $M\geq
0$.

As in Lemma 4.2, we define the following objects.

(4.27)
$\widehat{\Gamma}((x,0),M)=\left\\{\widehat{P}\in\mathcal{P}^{m+1}(\mathbb{R}^{n+d},\mathbb{R}):\,\begin{matrix}[l]\left|\partial^{\alpha}\widehat{P}(x,0)\right|\leq
M\text{ for }\alpha\in\mathbb{N}_{0}^{n+d},\,\left|\alpha\right|\leq m+1,\\\
\widehat{P}(\,\cdot\,,0)\equiv 0\text{, and
}\nabla_{v}\big{|}_{v=0}\widehat{P}\in\Gamma(x,M)\end{matrix}\right\\}.$
(4.28)
$\widehat{\Gamma}^{*}((x,0),M)=\left\\{\widehat{P}\in\mathcal{P}^{m+1}(\mathbb{R}^{n+d},\mathbb{R}):\,\begin{matrix}[l]\left|\partial^{\alpha}\widehat{P}(x,0)\right|\leq
M\text{ for }\alpha\in\mathbb{N}_{0}^{n+d},\,\left|\alpha\right|\leq m+1,\\\
\widehat{P}(\,\cdot\,,0)\equiv 0\text{, and
}\nabla_{v}\big{|}_{v=0}\widehat{P}\in\Gamma^{*}(x,M)\end{matrix}\right\\}.$

Recall that the first part of Theorem 1.6 was established in Lemma 3.1; thus
it suffices to prove the second part. By Taylor’s theorem and the definition
of Glaeser refinement, we see that if $(\Gamma(x))_{x\in E}$ has a section $F$
satisfying $\|F\|_{C^{m}(\mathbb{R}^{n},\mathbb{R}^{d})}\leq M$, then
$(\Gamma^{*}(x))_{x\in E}$ is nonempty and $\|\Gamma^{*}\|\leq CM$. Thus, it
suffices to suppose $(\Gamma^{*}(x))_{x\in E}$ is nonempty and determine the
existence of a section with the appropriate norm bounds.

Thanks to Lemma 4.2 and the assumption that $(\Gamma(x,M))_{x\in E,M\geq 0}$
is a closed regular $(C,1)$-shape field, we see that

(4.29) $(\widehat{\Gamma}((x,0),M))_{(x,0)\in\widehat{E},M\geq 0}$ is a closed
regular $(C,1)$-convex shape field.

We write

(4.30) $\widehat{\Gamma}((x,0))=\bigcup_{M\geq
0}\widehat{\Gamma}((x,0),M)\text{ and
}\widehat{\Gamma}^{*}((x,0))=\bigcup_{M\geq
0}\widehat{\Gamma}^{*}((x,0),M)\text{ for }(x,0)\in\widehat{E}.$

We will show that the hypothesis of Theorem 1.6 is satisfied for
$\widehat{E}$, $m+1$, $n+d$, $\widehat{\Gamma}$, and $\widehat{\Gamma}^{*}$ in
place of $E$, $m$, $n$, $\Gamma$, and $\Gamma^{*}$. In particular, we need to
show that

  1. (4.31)

$\widehat{\Gamma}^{*}((x,0))\neq\varnothing$ for each $(x,0)\in\widehat{E}$

  2. (4.33)

$\widehat{\Gamma}^{*}((x,0))$ is its own Glaeser refinement, and

  3. (4.35)

$\widehat{\Gamma}^{*}((x,0))\subset\widehat{\Gamma}((x,0))$ for $x\in E$.

Assuming the above, we see that the shape field
$(\widehat{\Gamma}((x,0),M))_{(x,0)\in\widehat{E},M\geq 0}$ satisfies the
hypotheses of Theorem 1.6 for $C^{m+1}(\mathbb{R}^{n+d},\mathbb{R})$. While we
did not show the iterated Glaeser refinement of $\widehat{\Gamma}$ terminates
in $\widehat{\Gamma}^{*}$, it is clear from definition that is must terminate
in a Glaeser stable bundle containing $\widehat{\Gamma}^{*}$; call this bundle
$\Gamma^{\prime}$. It follows that $\Gamma^{\prime}((x,0))$ is nonempty for
all $(x,0)\in\hat{E}$ from the fact that
$\widehat{\Gamma}((x,0))\subset\Gamma^{\prime}((x,0))$. The appropriate
quantitative bounds follow from
$\|\Gamma^{\prime}\|\leq\|\widehat{\Gamma}^{\prime}\|$.

Applying Theorem 1.6, there exists $G\in C^{m+1}(\mathbb{R}^{n+d},\mathbb{R})$
with

$\|G\|_{\dot{C}^{m+1}(\mathbb{R}^{n+d},\mathbb{R})}\leq\widehat{M}\text{ and
}J_{(x,0)}^{+}\in\widehat{\Gamma}^{*}((x,0),\widehat{M})\text{ for all
}(x,0)\in\widehat{E}.$

Here, $J_{(x,0)}^{+}$ denotes the $(m+1)$-jet at $(x,0)$, and
$\widehat{M}=C\|\Gamma^{\prime}\|\leq C\|\widehat{\Gamma}\|$ for some
controlled constant $C$. Thanks to the regularity condition ((2.1)), we can
further improve the control of the norm

(4.37) $\|G\|_{C^{m+1}(\mathbb{R}^{n+d},\mathbb{R})}\leq C\widehat{M}.$

Define

$\vec{F}(x)=\left(\partial_{v_{1}}G(x,0),\cdots,\partial_{v_{d}}G(x,0)\right).$

In view of the definition of $\widehat{\Gamma}$ in (4.28) and (4.30) and
(4.37), we have

(4.38) $\|\vec{F}\|_{\dot{C}^{m}(\mathbb{R}^{n},\mathbb{R}^{d})}\leq
C\widehat{M}\text{ and }J_{x}\vec{F}\in\Gamma^{*}(x,C\widehat{M}).$

In view of (4.38), we see that Lemma 4.1 holds.

It now suffices to establish ((4.31)), ((4.33)), and ((4.35)).

We show ((4.31)) as follows. Since we assume that
$\Gamma^{*}(x)\neq\varnothing$ for each $x\in E$, we can pick
$\vec{P}^{x}\in\Gamma^{*}(x)$ for each $x\in E$. We define

$\widehat{P}^{(x,0)}(y,v)=v\cdot\vec{P}^{x}(y)\in\mathcal{P}^{m}(\mathbb{R}^{n+d},\mathbb{R})\text{
for each }(x,0)\in\widehat{E}.$

We immediately verify that $\widehat{P}^{(x,0)}\in\widehat{\Gamma}^{*}((x,0))$
for each $(x,0)\in\widehat{E}$. Therefore,
$\widehat{\Gamma}^{*}((x,0))\neq\varnothing$ for each $(x,0)\in\widehat{E}$,
establishing ((4.31)).

Now, to show ((4.33)), fix $(x_{0},0)\in\widehat{E}$ and
$\widehat{P}_{0}\in\widehat{\Gamma}^{*}((x_{0},0))$. We will show that
$\widehat{P}_{0}$ survives the Glaeser refinement procedure. Let $\epsilon>0$.

Since $\widehat{P}_{0}\in\widehat{\Gamma}^{*}((x_{0},0))$, we can write

(4.39) $\widehat{P}_{0}(y,v)=\sum_{1\leq\left|\gamma\right|\leq
m+1}\frac{1}{\gamma!}v^{\gamma}P_{\gamma}(y)\text{ for some
}P_{\gamma}\in\mathcal{P}^{m}(\mathbb{R}^{n},\mathbb{R}).$

We set

(4.40)
$\begin{split}\vec{P}_{0}&=(P_{0,1},\cdots,P_{0,d})=\nabla_{v}\big{|}_{v=0}\widehat{P}_{0}\\\
&=\left(\partial_{v_{1}}\bigg{|}_{v=0}\left[\sum_{1\leq\left|\gamma\right|\leq
m+1}\frac{1}{\gamma!}v^{\gamma}P_{\gamma}(x)\right],\cdots,\partial_{v_{d}}\bigg{|}_{v=0}\left[\sum_{1\leq\left|\gamma\right|\leq
m+1}\frac{1}{\gamma!}v^{\gamma}P_{\gamma}(x)\right]\right).\end{split}$

By construction, we have

(4.41) $\vec{P}_{0}\in\Gamma^{*}(x_{0}).$

Thanks to (4.39) and (4.40), we can write

(4.42) $\widehat{P}_{0}(y,v)=\sum_{1\leq j\leq
d}v_{j}P_{0,j}(y)+\sum_{2\leq\left|\gamma\right|\leq
m+1}\frac{1}{\gamma!}v^{\gamma}P_{\gamma}(y).$

Since the bundle $(\Gamma^{*}(x))_{x\in E}$ is Glaeser stable, we know that
there exists $\delta>0$ such that for all $x_{1},\cdots,x_{{k^{\sharp}}}\in
E\cap B^{n}(x_{0},\delta)$, there exist

(4.43)
$\begin{split}\vec{P}_{1}&=(P_{1,1},\cdots,P_{1,d})\in\Gamma^{*}(x_{1}),\\\
&\vdots\\\
\vec{P}_{{k^{\sharp}}}&=(P_{{k^{\sharp}},1},\cdots,P_{{k^{\sharp}},d})\in\Gamma^{*}(x_{{k^{\sharp}}}),\end{split}$

with

(4.44)
$\left|\partial^{\alpha}(\vec{P}_{i}-\vec{P}_{j})(x_{i})\right|\leq\epsilon\left|x_{i}-x_{j}\right|^{m-\left|\alpha\right|}\text{
for }\left|\alpha\right|\leq m,0\leq i,j\leq{k^{\sharp}}.$

For any $(x_{1},0),\cdots,(x_{k^{\sharp}},0)\in\widehat{E}\cap
B^{n+d}((x_{0},0),\delta)$, we set

(4.45)
$\widehat{P}_{k}(y,v)=\sum_{j=1}^{d}v_{j}P_{k,j}(y)+\sum_{2\leq\left|\gamma\right|\leq
m+1}\frac{1}{\gamma!}v^{\gamma}P_{\gamma}(y)\text{ for }1\leq
k\leq{k^{\sharp}}.$

Here, $P_{\gamma}$ are as in (4.42), and
$\vec{P}_{1},\cdots,\vec{P}_{k^{\sharp}}$ are as in (4.43).

We claim that

(4.46) $\widehat{P}_{k}\in\widehat{\Gamma}((x_{k},0))\text{ for }1\leq
k\leq{k^{\sharp}}$

and

(4.47)
$\left|\partial^{\alpha}(\widehat{P}_{i}-\widehat{P}_{j})(x_{i},0)\right|\leq
C\epsilon\left|x_{i}-x_{j}\right|^{m+1-\left|\alpha\right|}\text{ for
}\left|\alpha\right|\leq m+1,0\leq i,j\leq{k^{\sharp}}.$

To verify (4.46), we apply $\nabla_{v}\big{|}_{v=0}$ to (4.45) and see that

(4.48)
$\nabla_{v}\big{|}_{v=0}\widehat{P}_{k}=(P_{k,1},\cdots,P_{k,d})=\vec{P}_{k}\in\Gamma^{*}(x_{k})\text{
for }1\leq k\leq{k^{\sharp}}.$

Therefor, (4.46) follows from (4.28) and (4.48).

To verify (4.47), we write
$\partial^{\alpha}=\partial_{x}^{\beta}\partial_{v}^{\xi}$. Observe that for
$0\leq k\leq{k^{\sharp}}$,

(4.49) $\partial_{v}^{\xi}\big{|}_{v=0}\widehat{P}_{k}=\begin{cases}0&\text{
if }\left|\xi\right|=0\\\ P_{k,j}\text{ for some $1\leq j\leq d$}&\text{ if
}\left|\xi\right|=1\\\ P_{\xi}&\text{ if }\left|\xi\right|\geq
2\end{cases}\,.$

Thanks to (4.49), we see that (4.47) holds trivially for $\left|\xi\right|\neq
1$. Therefore, it suffices to show (4.47) for the case $\left|\xi\right|=1$.
Without loss of generality, we may assume
$\partial_{v}^{\xi}=\partial_{v_{s}}$ for some
$s\in\left\\{1,\cdots,d\right\\}$. Then

(4.50)
$\begin{split}\left|\partial^{\alpha}(\widehat{P}_{i}-\widehat{P}_{j})(x_{i},0)\right|&=\left|\partial_{x}^{\beta}\partial_{v_{s}}(\widehat{P}_{i}-\widehat{P}_{j})(x_{i},0)\right|\\\
&=\left|\partial_{x}^{\beta}(P_{i,s}-P_{j,s})(x_{i})\right|\\\
&\leq\epsilon\left|x_{i}-x_{j}\right|^{m-\left|\beta\right|}\quad\text{(thanks
to \eqref{eq.red-4})}\\\
&=\epsilon\left|x_{i}-x_{j}\right|^{m+1-\left|\alpha\right|}.\end{split}$

We see that (4.47) follows from (4.50). We have shown that the bundle
$(\widehat{\Gamma}^{*}((x,0)))_{(x,0)\in\widehat{E}}$ as defined in (4.28) and
(4.30) is Glaeser stable, giving us ((4.33)).

Lastly, we get ((4.35)) by observing that
$\widehat{\Gamma}^{*}((x,0))\subset\widehat{\Gamma}((x,0))$ since
$\Gamma^{*}((x,0))\subset\Gamma((x,0))$ for all $(x,0)\in\hat{E}$.

∎

######  Remark 4.1.

Thanks to Lemma 4.1, to prove Theorem 1.6 for
$C^{m}(\mathbb{R}^{n},\mathbb{R}^{d})$, it suffices to prove it for the case
$d=1$.

For the rest of the paper, we write $C^{m}(\mathbb{R}^{n})$ instead of
$C^{m}(\mathbb{R}^{n},\mathbb{R})$, and $F$, $\mathcal{P}$, $P$ instead of
$\vec{F}$, $\vec{\mathcal{P}}$, $\vec{P}$.

##  5\. Finiteness principle for shape fields

Recall from the previous section that we will be working with real-valued
functions for the rest of the paper.

######  Definition 5.1.

A function $\omega:[0,1]\to[0,\infty)$ is called a regular modulus of
continuity if it satisfies the following conditions:

  1. ($\omega$-1)

$\omega(0)=\lim\limits_{t\downarrow 0}\omega(t)=0$ and $\omega(1)=1$;

  2. ($\omega$-1)

$\omega(t)$ is increasing on $[0,1]$; and

  3. ($\omega$-1)

$\omega(t)/t$ is decreasing on $(0,1]$.

Note that in ($\omega$-2) and ($\omega$-3), we do not demand that $\omega$ be
strictly increasing, or that $\omega(t)/t$ be strictly decreasing.

Let $\omega$ be a regular modulus of continuity. We write
$C^{m,\omega}(\mathbb{R}^{n})$ to denote the space of all $C^{m}$ functions
$F$ on $\mathbb{R}^{n}$ for which the norm

$\|F\|_{C^{m,\omega}(\mathbb{R}^{n})}:=\sup_{\begin{subarray}{c}x\in\mathbb{R}^{n}\\\
\left|\alpha\right|\leq
m\end{subarray}}\left|\partial^{\alpha}F(x)\right|+\sup_{\begin{subarray}{c}\left|\alpha\right|=m\\\
x,x^{\prime}\in\mathbb{R}^{n}\\\ 0<\left|x-x^{\prime}\right|\leq
1\end{subarray}}\frac{\left|\partial^{\alpha}F(x)-\partial^{\alpha}F(x^{\prime})\right|}{\omega(\left|x-x^{\prime}\right|)}$

is finite.

Similarly, we write $\dot{C}^{m,\omega}(\mathbb{R}^{n})$ to denote the space
of all $C^{m}$ functions $F$ on $\mathbb{R}^{n}$ for which the norm

$\|F\|_{C^{m,\omega}(\mathbb{R}^{n})}:=\sup_{\begin{subarray}{c}\left|\alpha\right|=m\\\
x,x^{\prime}\in\mathbb{R}^{n}\\\ 0<\left|x-x^{\prime}\right|\leq
1\end{subarray}}\frac{\left|\partial^{\alpha}F(x)-\partial^{\alpha}F(x^{\prime})\right|}{\omega(\left|x-x^{\prime}\right|)}$

is finite.

The following lemma was proven in [9].

######  Lemma 5.1.

Let $(\Gamma(x,M))_{x\in E,M\geq 0}$ be a $(C_{w},\delta_{\max})$-convex shape
field. Let $\delta\in(0,\delta_{\max})$, $x\in E$, $M\geq 0$,
$A^{\prime},A^{\prime\prime}>0$,
$P_{1},\cdots,P_{k},Q_{1},\cdots,Q_{k}\in\mathcal{P}$. Assume that

  * •

$P_{i}\in\Gamma(x,A^{\prime}M)$ for $i=1,\cdots,k$;

  * •

$\left|\partial^{\alpha}(P_{i}-P_{j})(x)\right|\leq
A^{\prime}M\delta^{m-\left|\alpha\right|}$ for $\left|\alpha\right|\leq m$,
$1\leq i,j\leq k$;

  * •

$\left|\partial^{\alpha}Q_{i}(x)\right|\leq
A^{\prime\prime}\delta^{-\left|\alpha\right|}$ for $\left|\beta\right|\leq m$
and $1\leq i\leq k$;

  * •

$\sum_{i=1}^{k}Q_{i}\odot_{x}Q_{i}=1$.

Then $\sum_{i=1}^{k}Q_{i}\odot_{x}Q_{i}\odot_{x}P_{i}\in\Gamma(x,CM)$, with
$C=C(A^{\prime},A^{\prime\prime},C_{w},m,n,k)$.

By adapting the proof of the Finiteness Principle for shape fields from [9]
(see also [3, 5]), we obtain the following result.

######  Theorem 5.2 ($\dot{C}^{m,\omega}$-Finiteness Principle for Shape
Fields).

Let $m,n\in\mathbb{N}$. For sufficiently large
$k^{\sharp}_{\mathrm{SF}}=k(m,n)$, the following holds.

Let $E\subset\mathbb{R}^{n}$ be an arbitrary subset and $\omega$ be a regular
modulus of continuity. Let $\Gamma(x,M)_{x\in E,M\geq 0}$ be a
$(C_{w},\delta_{\max})$-convex shape field. Let $M_{0}<\infty$ and $Q_{0}$ be
a cube of sidelength $\delta_{Q_{0}}\leq\delta_{\max}$. Assume that for each
$S\subset E$ with $\\#(S)\leq{k^{\sharp}}_{SF}$, there exists $(P^{x})_{x\in
S}$ such that

(5.1) $P^{x}\in\Gamma(x,M_{0})\text{ for }x\in S$

and

(5.2) $|\partial^{\alpha}(P^{x}-P^{y})(x)|\leq
M_{0}\omega(|x-y|)|x-y|^{m-|\alpha|}\text{ for }|\alpha|\leq m,x,y\in S,x\neq
y.$

Then there exists $F\in C^{m,\omega}(Q_{0})$, with

(5.3) $J_{x}F\in\Gamma(x,C^{\sharp}M_{0})\text{ for }x\in E\cap Q_{0}$

and

(5.4) $\|F\|_{\dot{C}^{m,\omega}(Q_{0})}\leq C^{\sharp}M_{0}.$

Here, $C^{\sharp}=C^{\sharp}(m,n,C_{w})$.

##  6\. Other Preliminaries

######  Lemma 6.1 (Finiteness Lemma).

Let $E\subset\mathbb{R}^{n}$ be a compact set. Let $(\Gamma(x,M))_{x\in
E,M\geq 0}$ be a regular Glaeser stable shape field. Then there exists a
finite constant $A^{\sharp}$ such that:

Given $x_{1},...,x_{k^{\sharp}}\in E$, there exist
$P_{i}\in\Gamma(x_{i},A^{\sharp})$ such that

(6.1) $|\partial^{\alpha}(P_{i}-P_{j})(x_{j})|\leq
A^{\sharp}|x_{i}-x_{j}|^{m-|\alpha|}\text{ for }|\alpha|\leq m,1\leq i,j\leq
k^{\sharp}.$

The $A^{\sharp}$ from Lemma 6.1 will be fixed and referenced throughout the
remainder of the paper.

###### Proof.

We slightly modify the proof given in [6].

Suppose towards a contradiction, that for each $\nu=1,2,\cdots$ we can find

  * •

$x^{(\nu)}_{1},\cdots,x^{(\nu)}_{k^{\sharp}}\in E$, and

  * •

$A^{(\nu)}>0$, with $\lim\limits_{\nu\to\infty}A^{(\nu)}=\infty$,

such that for each $\nu$,

  * (6.2)

there do not exist polynomials $P_{1},\cdots,P_{k^{\sharp}}\in\mathcal{P}$
such that

(6.4) $P_{j}\in\Gamma(x_{j}^{(\nu)},A^{(\nu)})\text{ for
}j=1,\cdots,{k^{\sharp}}\text{; and}$ (6.5)
$\left|\partial^{\alpha}(P_{i}-P_{j})(x^{(\nu)}_{j})\right|\leq
A^{(\nu)}\left|x^{(\nu)}_{i}-x^{(\nu)}_{j}\right|^{{m-\left|\alpha\right|}}\text{
for }\left|\alpha\right|\leq m,0\leq i,j\leq{k^{\sharp}}.$

Since $E$ is compact, by passing to a subsequence, we may assume that

$x^{(\nu)}_{j}\to x^{(\infty)}_{j}\in E\text{ as }\nu\to\infty,\text{ for
}j=1,\cdots,{k^{\sharp}}.$

Let $z_{1},\cdots,z_{\mu_{\max}}$ be an enumeration of the distinct elements
of $\left\\{x^{(\infty)}_{1},\cdots,x^{(\infty)}_{k^{\sharp}}\right\\}$. For
each $\mu=1,\cdots,{\mu_{\max}}$, let
$S(\mu):=\left\\{j:x^{(\infty)}_{j}=z_{\mu},\,1\leq
j\leq{k^{\sharp}}\right\\}$. Therefore, for sufficiently large $\nu$, we have

(6.6) $\left|x^{(\nu)}_{j}-z_{\mu}\right|\leq\eta_{1}\text{ for all }j\in
S(\mu),$

and

(6.7) $\left|x^{(\nu)}_{j}-x^{(\nu)}_{j^{\prime}}\right|\geq\eta_{2}\text{ for
}j\in S(\mu),\,j^{\prime}\in S(\mu^{\prime})\text{ with }\mu\neq\mu^{\prime},$

where $\eta_{1}$ and $\eta_{2}>0$ are chosen to be independent of $\nu$. For
the rest of the proof, we use $A_{0}$, $A_{1}$, etc., to denote constants
independent of $\nu$.

We now apply the hypothesis that $(\Gamma(x))_{x\in E}$ is Glaeser-stable.

Fix $\mu\in\left\\{1,\cdots,{\mu_{\max}}\right\\}$. We set $x_{0}=z_{\mu}$,
$P_{0}\in\Gamma(x_{0},A_{0})$, and $\epsilon=1$ in the definition of Glaeser
refinement. Let $\nu$ be sufficiently large. We set $x_{j}:=x^{(\nu)}_{j}$ for
$j\in S(\mu)$, $x_{j}:=z_{\mu}$ for $j\notin S(\mu)$ ($1\leq
j\leq{k^{\sharp}}$). Since $\Gamma(x)$ is its own Glaeser refinement, it
follows from (6.6) that there exist

(6.8) $P^{(\nu)}_{j}\in\Gamma(x^{(\nu)}_{j})\text{, for }j\in S(\mu)$

$\nu$ sufficiently large, such that

(6.9)
$\left|\partial^{\alpha}(P^{(\nu)}_{i}-P^{(\nu)}_{j})(x^{(\nu)}_{j})\right|\leq\left|x^{(\nu)}_{i}-x^{(\nu)}_{j}\right|^{m-\left|\alpha\right|}\text{
for }i,j\in S(\mu),\left|\alpha\right|\leq m.$

Furthermore, it follows from (6.6), (6.8), (6.9), and ((2.3)) (upon taking
$\eta_{1}$ sufficiently small) that

(6.10) $P^{(\nu)}_{j}\in\Gamma(x^{(\nu)}_{j},A_{1})\text{, for }j\in S(\mu).$

We repeat the argument above for $\mu=1,\cdots,{\mu_{\max}}$. Thus, for
sufficiently large $\nu$, we can find polynomials
$P^{(\nu)}_{1},\cdots,P^{(\nu)}_{k^{\sharp}}$ that satisfy the following:

(6.11) $P^{(\nu)}_{j}\in\Gamma(x^{(\nu)}_{j},A_{2})\text{ for
}j=1,\cdots,{k^{\sharp}};\text{ and }$ (6.12)
$\left|\partial^{\alpha}(P^{(\nu)}_{i}-P^{(\nu)}_{j})(x^{(\nu)}_{j})\right|\leq\left|x^{(\nu)}_{i}-x^{(\nu)}_{j}\right|^{{m-\left|\alpha\right|}}\text{
for }\left|\alpha\right|\leq m,\,i,j\in S(\mu),\,\mu=1,\cdots,{\mu_{\max}}.$

Observe that by (2.1), (6.11) implies

(6.13) $|\partial^{\alpha}P_{j}^{(\nu)}(x_{j}^{(\nu)}|\leq A_{2}\text{ for
}|\alpha|\leq m.$

Combining (6.7) and (6.13), we see that

(6.14)
$\left|\partial^{\alpha}(P^{(\nu)}_{i}-P^{(\nu)}_{j})(x^{(\nu)}_{j})\right|\leq
A_{3}\left|x^{(\nu)}_{i}-x^{(\nu)}_{j}\right|^{{m-\left|\alpha\right|}}\text{
for }\left|\alpha\right|\leq m,\,i\in S(\mu),\,j\in
S(\mu^{\prime}),\,\mu\neq\mu^{\prime}.$

Together with (6.12) and (6.14), we see that

(6.15)
$\left|\partial^{\alpha}(P^{(\nu)}_{i}-P^{(\nu)}_{j})(x^{(\nu)}_{j})\right|\leq
A_{4}\left|x^{(\nu)}_{i}-x^{(\nu)}_{j}\right|^{m-\left|\alpha\right|}\text{
for }\left|\alpha\right|\leq m,\,1\leq i,j\leq{k^{\sharp}}.$

For sufficiently large $\nu$, we have $A^{(\nu)}>A_{2}$ and $A^{(\nu)}>A_{4}$,
with $A_{2}$ and $A_{4}$, respectively, as in and (6.11) (6.15). Then, (6.11),
and (6.15) together contradict ($\star$).

The lemma is proved.

∎

######  Lemma 6.2.

[6] Let $S\subset\mathbb{R}^{n}$ with $2\leq\\#(S)\leq{k^{\sharp}}$. Then we
may partition $S$ into subsets $S_{1},\cdots,S_{{\nu_{\max}}}$ with the
following properties.

  1. (a)

$\\#(S_{\ell})\leq\\#(S)$ for each $\nu=1,\cdots,{\nu_{\max}}$.

  2. (b)

If $x\in S_{\nu}$ and $y\in S_{\mu}$ with $\nu\neq\mu$, then
$\left|x-y\right|>c({k^{\sharp}})\cdot{\rm diam}(S)$. Here, $c({k^{\sharp}})$
depends only on ${k^{\sharp}}$.

######  Lemma 6.3.

Suppose $W\subset\mathbb{R}^{n}$ is an $r$-dimensional subspace, $w\in W$, and
$w_{0},...,w_{r}$ form a nondegenerate affine $r$-simplex containing
$B(w,\delta)\cap W$ for some $\delta>0$. Then, there exists $\eta>0$ for which
the following holds:

Let $W^{\prime}\subset\mathbb{R}^{n}$ be another $r$-dimensional subspace. If
$w_{0}^{\prime},...,w_{r}^{\prime}\in W^{\prime}$ satisfy

(6.16) $|w_{j}-w_{j}^{\prime}|<\eta\text{ for }j=0,1,\cdots,r,$

and $w^{\prime}\in W^{\prime}$ satisfies

(6.17) $|w-w^{\prime}|<\eta,$

then $w^{\prime}$ is contained in the convex hull of
$w_{0}^{\prime},...,w_{r}^{\prime}$.

###### Proof.

Since $w\in Conv(w_{0},w_{1},\cdots,w_{r})$, there exist
$\lambda_{0},\lambda_{1},\cdots,\lambda_{r}$ such that
$w=\sum_{i=0}^{r}\lambda_{i}w_{i}$. Let $\xi_{1},\cdots,\xi_{r}$ be an
orthonormal basis for $\mathrm{span}(w_{0}-w_{1},\cdots,w_{0}-w_{r})$. Then
the coefficients $\lambda_{0},\lambda_{1},\cdots,\lambda_{r}$ satisfy the
following constrained linear system.

(6.18)
$\displaystyle\sum_{i=0}^{r}\lambda_{i}(w_{i}\cdot\xi_{j})=w\cdot\xi_{j}\text{
for }j=1,\cdots,r;$ (6.19) $\displaystyle\sum_{i=0}^{r}\lambda_{i}=1;$ (6.20)
$\displaystyle 0\leq\lambda_{i}\leq 1\text{ for }i=0,1,\cdots,r.$

Note that the system (6.18)–(6.19) is nondegenerate, since
$w_{0},w_{1},\cdots,w_{r}$ form a nondegenerate affine $r$-simplex.

Moreover, since $B(w,\delta)\cap W\subset Conv(w_{0},w_{1},\cdots,w_{r})$, we
may replace (6.20) by

(6.21) $c\leq\lambda_{i}\leq 1\text{ for }i=0,1,\cdots,r$

for some small constant $c>0$.

If $\eta$ is sufficiently small, (6.16) implies that
$w_{0}^{\prime},\cdots,w_{r}^{\prime}$ also form a nondegenerate $r$-simplex
in $W^{\prime}$. We may find
$\lambda_{0}^{\prime},\lambda_{1}^{\prime},\cdots,\lambda_{r}^{\prime}$ such
that

(6.22)
$\displaystyle\sum_{i=0}^{r}\lambda_{i}^{\prime}(w_{i}^{\prime}\cdot\xi_{j})=w^{\prime}\cdot\xi_{j}\text{
for }j=1,\cdots,r;$ (6.23)
$\displaystyle\sum_{i=0}^{r}\lambda_{i}^{\prime}=1.$

Note that $w^{\prime}\in Conv(w_{0}^{\prime},\cdots,w_{r}^{\prime})$ if and
only if
$\lambda_{0}^{\prime},\lambda_{1}^{\prime},\cdots,\lambda_{r}^{\prime}$
satisfy the system (6.22) and (6.23) coupled with the constraint

(6.24) $0\leq\lambda_{i}^{\prime}\leq 1\text{ for }i=0,1,\cdots,r.$

By choosing $\eta$ to be sufficiently small, (6.17) and (6.16) imply that the
coefficients of the linear system (6.18)–(6.19) are sufficiently close to that
of (6.22)–(6.23). This forces $\lambda_{i}$ to be sufficiently close to to
$\lambda_{i}^{\prime}$ for $i=0,1,\cdots,r$. Thanks to (6.21), we see that
$\lambda_{0}^{\prime},\lambda_{1}^{\prime},\cdots,\lambda_{r}^{\prime}$
satisfy the constraint (6.24). This proves the lemma. ∎

######  Lemma 6.4 (Helly’s Theorem).

Let $(\mathcal{K}_{\alpha})_{\alpha\in\mathcal{A}}$ be a family of compact,
convex sets in $\mathbb{R}^{N}$. If any $N+1$ of the $\mathcal{K}_{\alpha}$
have nonempty intersection, then
$\bigcap_{\alpha\in\mathcal{A}}\mathcal{K}_{\alpha}\neq\emptyset$.

For a proof of the above, see [24], for example.

######  Lemma 6.5 (Whitney Extension Theorem for Finite Sets).

[32] Let $S\subset\mathbb{R}^{n}$ be finite, and for each $x\in S$, let
$P^{x}\in\mathcal{P}$. If

(6.25) $|\partial^{\alpha}P^{x}(x)|\leq M\text{ for }|\alpha|\leq m$

and

(6.26) $|\partial^{\alpha}(P^{x}-P^{y})(x)|\leq M|x-y|^{m-|\alpha|}\text{ for
}|\alpha|\leq m,x,y\in S,x\neq y,$

then there exists $F\in C^{m}(\mathbb{R}^{n})$ such that
$\|F\|_{C^{m}(\mathbb{R}^{n})}\leq CM$ and $J_{x}F=P^{x}$ for all $x\in S$.
Here, $C$ depends only on $m,n$.

##  7\. Convex Sets and Strata

In this section, we write $A,A^{\prime},$ etc., to denote quantities that are
strictly greater than $A^{\sharp}$, with $A^{\sharp}$ as in Lemma 6.1.

######  Definition 7.1.

For $x_{0}\in E$, $k\in\mathbb{N}_{0}$, and $A>0$, we let $\Gamma(x_{0},k,A)$
be the set of $P_{0}\in\Gamma(x)$ such that for any $x_{1},...,x_{k}\in E$,
there exist $P_{1},\cdots,P_{k}\in\mathcal{P}$, with

  * •

$P_{i}\in\Gamma(x,A)$ for $i=1,\cdots,k$; and

  * •

$|\partial^{\alpha}(P_{i}-P_{j})(x_{j})|\leq A|x_{i}-x_{j}|^{m-|\alpha|}\text{
for }|\alpha|\leq m,0\leq i,j\leq k$.

Note that by the Finiteness Lemma, there exists $A^{\sharp}<\infty$ such that
$\Gamma(x,k,A^{\sharp})$ is nonempty for all $x\in E$ and $k\leq{k^{\sharp}}$.

Let $\overline{\mathcal{R}}_{x}$ denote the ring of $(m-1)$-jets of functions
at $x$ and let $\pi_{x}:\mathcal{R}_{x}\to\overline{\mathcal{R}}_{x}$ be the
natural projection. We identify $\overline{\mathcal{R}}_{x}$ with the space
$\overline{\mathcal{P}}$ of polynomials with degree no greater than $(m-1)$.
For $A>0$, we define

(7.1) $\overline{\Gamma}(x,k,A):=\pi_{x}\Gamma(x,k,A).$

We note that $\overline{\Gamma}(x,k,A)$ is convex, since $\pi_{x}$ is linear.

Similarly, let

(7.2) $\overline{\Gamma}(x,A):=\pi_{x}\Gamma(x,A).$

###  7.1. Strata

The following three lemmas will assist in defining the dimension of a Glaeser
stable bundle.

######  Lemma 7.1.

Let $A\geq 2A^{\sharp}$, $x_{0}\in E$, and $k\in\mathbb{N}$. Then,

  1. (1)

$\dim\Gamma(x_{0})=\dim\Gamma(x_{0},A)=\dim\Gamma(x_{0},2A^{\sharp})$.

  2. (2)

$\dim\Gamma(x_{0},k,A)=\dim\Gamma(x_{0},k,2A^{\sharp})$.

###### Proof.

Suppose the contrary to the first conclusion, that there exists
$A>2A^{\sharp}$ such that
$\dim\Gamma(x_{0},A)>\dim\Gamma(x_{0},2A^{\sharp})=d$ and pick
$P\in\Gamma(x_{0},A)$ such that $P$ is not contained in the $d$-dimensional
hyperplane in $\mathcal{P}$ containing $\Gamma(x_{0},2A^{\sharp})$.

By Lemma 6.1, $\Gamma(x_{0},A^{\sharp})$ is nonempty, so take
$P^{\prime}\in\Gamma(x_{0},A^{\sharp})$.

Now consider the sequence $P_{n}=(1/n)P^{\prime}+(1-1/n)P$. By convexity,
$P_{n}\in\Gamma(x_{0},A)$ for all $n$. But by supposition,
$P_{n}\notin\Gamma(x_{0},2A^{\sharp})$ for all $n$, contradicting ((2.5)).
Thus, $\dim\Gamma(x_{0},A)=\dim\Gamma(x_{0},2A^{\sharp})$.
$\dim\Gamma(x_{0})=\dim\Gamma(x_{0},2A^{\sharp})$ follows by taking unions.

Now suppose the contrary to the second conclusion. Analogous to the previous
situation, there exists $A>2A^{\sharp}$ such that
$\dim\Gamma(x_{0},k,A)>\dim\Gamma(x_{0},k,2A^{\sharp})=d^{\prime}$ and pick
$P\in\Gamma(x_{0},k,A)$ such that $P$ is not contained in the
$d^{\prime}$-dimensional hyperplane in $\mathcal{P}$ containing
$\Gamma(x_{0},k,2A^{\sharp})$.

Let $P^{\prime}\in\Gamma(x_{0},k,A^{\sharp})$ and again consider the sequence
$P_{n}=(1/n)P^{\prime}+(1-1/n)P$. By our prior reasoning, there exists $N$
such that $n\geq N$ implies $P_{n}\in\Gamma(x_{0},3/2A^{\sharp})$.

Now, let $x_{1},...,x_{{\overline{k}}}\in E$. Choose $Q_{j}\in\Gamma(x_{j},1)$
such that

(7.3)
$|\partial^{\alpha}(Q_{i}-Q_{j})(x_{i})|\leq|x_{i}-x_{j}|^{m-|\alpha|}\text{
for }|\alpha|\leq m,0\leq i,j\leq k$

with $Q_{0}=P$. Similarly, choose $Q_{j}^{\prime}\in\Gamma(x_{j},A)$ such that

(7.4) $|\partial^{\alpha}(Q^{\prime}_{i}-Q^{\prime}_{j})(x_{i})|\leq
A|x_{i}-x_{j}|^{m-|\alpha|}\text{ for }|\alpha|\leq m,0\leq i,j\leq k$

with $Q_{0}^{\prime}=P^{\prime}$.

Form sequences $Q_{j}^{(n)}=(1/n)Q_{j}^{\prime}+(1-1/n)Q_{j}$. By ((2.5)) (and
the independence of $\delta$ of $x$ in this condition), we may take $N$
independent of $x_{1},...,x_{k}$ such that $n\geq N$ implies
$Q_{j}^{(n)}\in\Gamma(x_{j},2A^{\sharp})$. Thus, for $n\in N$,
$P^{\prime}_{n}\in\Gamma(x_{0},k,2A^{\sharp})$. This is a contradiction, so
$\dim\Gamma(x_{0},k,A)=\dim\Gamma(x_{0},k,2A^{\sharp})$. ∎

######  Lemma 7.2.

Let $A\geq 2A^{\sharp}$, $x_{0}\in E$, and $k\leq{k^{\sharp}}$. Then,

(7.5) $\dim\Gamma(x_{0},A)=\dim\Gamma(x_{0},k,A).$

###### Proof.

Suppose the contrary. Then by Lemma 7.1,
$\dim\Gamma(x_{0},k,A^{\prime})\leq\dim\Gamma(x_{0},A)-1$ for all
$A^{\prime}>A$. In particular, there exists $P\in\Gamma(x_{0},A)$ such that
for all $A^{\prime}>0$ $P\notin\Gamma(x_{0},k,A^{\prime})$.

Define the convex bundle $(\Gamma^{\prime}(x))_{x\in E}$, where

(7.6) $\Gamma^{\prime}(x,M)=\left\\{\begin{array}[]{lr}\Gamma(x,M)&\text{if
}M\geq 0,x\neq x_{0}\\\ \\{P\\}&\text{if }x=x_{0},M\geq A\\\
\emptyset&\text{if }x=x_{0},M<A.\end{array}\right.$

Observe that

By Lemma 6.1 and the fact $k\leq{k^{\sharp}}$, there exists $\tilde{A}>0$ such
that $\Gamma^{\prime}(x_{0},k,\tilde{A})$ is nonempty (in particular,
$\Gamma^{\prime}(x_{0},k,\tilde{A})=\\{P\\}$), where the
$\Gamma^{\prime}(x,k,A)$ are defined analogously to the original
$\Gamma(x,k,A)$ but with $\Gamma^{\prime}(x,M)$ in place of $\Gamma(x,M)$.

Since $\Gamma^{\prime}(x,A)\subset\Gamma(x,A)$ for all $x\in E$ and $A>0$,
$\Gamma^{\prime}(x_{0},k,\tilde{A})\subset\Gamma(x_{0},k,\tilde{A})$ for all
$x\in E,A>0$, thus $P\in\Gamma(x_{0},k,\tilde{A})$, a contradiction. ∎

Motivated by the prior two lemmas, we make the following definition. Given a
graded holding space $(H(x,M))_{x\in E}$, define $W_{H}(x)$ as the (unique)
$\dim H(x)$-dimensional subspace of $\mathcal{P}$ containing $H(x)$.

######  Lemma 7.3.

Let $A\geq 2A^{\sharp},x\in E,$ and $k\leq{k^{\sharp}}$. Then,

(7.7) $\dim
W_{\Gamma}(x)=\dim\Gamma(x)=\dim\Gamma(x,A)=\dim\Gamma(x,k,A)=\dim\Gamma(x,2A^{\sharp})=\dim\Gamma(x,k,2A^{\sharp}).$

and

(7.8)
$\dim\pi_{x}W_{\Gamma}(x)=\dim\overline{\Gamma}(x)=\dim\overline{\Gamma}(x,A)=\dim\overline{\Gamma}(x,k,A)=\dim\overline{\Gamma}(x,2A^{\sharp})=\dim\overline{\Gamma}(x,k,2A^{\sharp}).$

###### Proof.

Lemmas 7.1 and 7.2 immediately imply (7.7). Since each of the relevant
$\Gamma$’s is contained in $\Gamma(x)$, not only are they all the same
dimension, but they are contained in the same $\dim\Gamma(x)$-dimensional
hyperplane. Thus, the dimensions of the projections under $\pi_{x}$ are all
equal as well, establishing (7.8). ∎

######  Definition 7.2.

Let $E\subset\mathbb{R}^{n}$ be a compact set. Let $(H(x))_{x\in E}$ be a
convex bundle over $E$. For each $x\in E$, we define the signature of $x$ to
be

${\rm sig}(x):=\left(\dim H(x),\dim[\ker\pi_{x}\cap H(x)]\right).$

For given integers $k_{1},k_{2}$, we define

$E(k_{1},k_{2}):=\\{x\in E:{\rm sig}(x)=(k_{1},k_{2})\\}.$

We define the stratum

$E_{1}:=E(k_{1}^{*},k_{2}^{*})$

where $k_{1}^{*}$ is as small as possible, and $k_{2}^{*}$ is as large as
possible for this given $k_{1}^{*}$. We call $E_{1}$ the lowest stratum.

###  7.2. Lowest stratum

######  Lemma 7.4.

Let $E\subset\mathbb{R}^{n}$ and $(H(x))_{x\in E}$ be a convex, Glaeser stable
bundle. Let $E_{1}$ be the lowest stratum as in Definition 7.2. Then $E_{1}$
is compact.

###### Proof.

Let $x\in E$. Suppose $\dim H(x)=d$. Let $P_{1},\cdots,P_{d}$ be the vertices
of a nondegenerate affine $d$-simplex in $\text{int}H(x)$. Thus, small
perturbation of $\\{P_{1},\cdots,P_{d}\\}$ remain the vertices of a
nondegenerate affine $d$-simplex.

By the Glaeser stability of $(H(x))_{x\in E}$, for any $\tilde{x}\in E$
sufficiently close to $x$, we may find $\tilde{P}_{1},\cdots,\tilde{P}_{d}\in
H(\tilde{x})$, such that they are the vertices of a nondegenerate affine
$d$-simplex. Hence,

$\dim(H(\tilde{x}))\geq d\text{ for any $\tilde{x}$ sufficiently close to
$x$.}$

It follows that

$\\{x\in E:\dim H(x)<d\\}$

is a closed set for any $d\in\mathbb{N}_{0}$. In particular, for
$k_{1}=\min_{y\in E}\dim H(y)$, the set

(7.9) $E(k_{1}):=\bigcup_{k_{2}\in\mathbb{N}_{0}}E(k_{1},k_{2})\text{ is
closed.}$

By hypothesis of Theorem 1.6 again, we see that the map

$x\mapsto W_{H}(x)$

is continuous from $E(k_{1})$ to $\mathcal{G}$, the Grassmannian of
$k_{1}$-planes in $\mathcal{P}$.

Set $k_{2}=\max_{y\in E(k_{1})}\dim(\ker\pi_{y}\cap W_{H}(y))$, noting that
$\dim(\ker\pi_{y}\cap W_{H}(y))=\dim(\ker\pi_{y}\cap H(y))$. By Definition
7.2, we see that

$E_{1}=\\{x\in E(k_{1},k_{2}):\dim(\ker\pi_{x}\cap H(x))=k_{2}\\}.$

We will show that $E_{1}$ is closed.

Let $(x_{\nu})_{\nu=1}^{\infty}$ be a sequence in $E_{1}$ converging to
$x\in\mathbb{R}^{n}$. By (7.9), we know that $x\in E(k_{1})$, and thus,
$W_{H}(x_{\nu})$ converges to $W_{H}(x)$ in $\mathcal{G}$. Passing onto a
subsequence, we can further assume that $\ker\pi_{x_{\nu}}\cap W_{H}(x_{\nu})$
converges to some (not necessarily unique) $W\in Conv(\mathcal{P})$ with $\dim
W=k_{2}$.

Now, $W\in H(x)$ and $\pi_{x}|_{W}\equiv 0$. Therefore,

$\dim(\ker\pi_{x}\cap W_{H}(x))\geq k_{2}.$

By our choice of $k_{2}$, we must have

$\dim(\ker\pi_{x}\cap W_{H}(x))=k_{2}.$

Hence, $x\in E_{1}$ as claimed.

Since $E$ is compact and $E_{1}\subset E$ is closed, the lemma follows.

∎

Recall from Definition 7.2 that $E_{1}$ denotes the lowest stratum of $E$. By
Lemma 7.4, $E_{1}$ is compact. Furthermore, for a convex bundle $(H(x))_{x\in
E}$, $\dim H(x)$ and $\dim(\ker\pi_{x}\cap H(x))$ are constant for all $x\in
E_{1}$.

Limiting our attention to the bundle $(\Gamma(x,M))_{x\in E}$ from the
hypotheses of Theorem 1.6, we set

$\begin{split}d&:=\dim\Gamma(x)\text{ for all }x\in E_{1},\text{ and }\\\
{\bar{d}}&:=\dim\overline{\Gamma}(x)\text{ for all }x\in E_{1}.\end{split}$

###  7.3. Glaeser Stability

Given a shape field $(\Gamma(x,M))_{x\in E,M\geq 0},x_{0}\in E,$ and $A>0$,
define

(7.10) $\Gamma^{o}(x,A):=\bigcup_{M<A}\Gamma(x,M).$

######  Lemma 7.5.

Let $(\Gamma(x,M))_{x\in E}$ be a regular, convex, Glaeser-stable, shape field
and fix $A>0$. The bundle $(\Gamma^{o}(x,A))_{x\in E}$ is Glaeser stable.

###### Proof.

Fix $x_{0}\in E$ and $P_{0}\in\Gamma^{o}(x,A)$. Then, there exists $M<A$ such
that $P_{0}\in\Gamma(x,M)$. By (2.3), there exists $\eta>0$ such that for any
$x^{\prime}\in E\cap B(x,\eta)$, if $P^{\prime}\in\Gamma(x^{\prime})$ with

(7.11)
$|\partial^{\alpha}(P-P^{\prime})(x)|,|\partial^{\alpha}(P-P^{\prime})(x^{\prime})|\leq\eta|x-x^{\prime}|^{m-|\alpha|}\text{
for all }|\alpha|\leq m,$

then
$P^{\prime}\in\Gamma(x^{\prime},\frac{M+A}{2})\subset\Gamma^{o}(x^{\prime},A)$.

Let $0<\epsilon<\eta$. By the Glaeser stability of $(\Gamma(x))_{x\in E}$,
there exists $0<\delta<\eta$ such that if $x_{1},...,x_{{k^{\sharp}}}\in
E\cap(x,\delta)$, then there exist $P_{1},...,P_{{k^{\sharp}}}\in\mathcal{P}$
such that

(7.12) $P_{j}\in\Gamma(x_{j})\text{ for }j=1,...,{k^{\sharp}}$

and

(7.13)
$|\partial^{\alpha}(P_{i}-P_{j})(x_{i})|<\eta|x_{i}-x_{j}|^{m-|\alpha|}\text{
for }|\alpha|\leq m,0\leq i,j\leq{k^{\sharp}}.$

Since (7.13) implies (7.11), we have $P_{j}\in\Gamma^{o}(x_{j},A)$ for
$j=1,...,{k^{\sharp}}$, which is precisely what we wanted to show.

∎

In the sequel, we use $\Gamma(x,A)$ to denote $\Gamma^{o}(x,A)$, simplifying
notation. Note that $\Gamma^{o}(x)$ may not be a closed subset of
$\mathcal{P}$, but since
$\Gamma^{o}(x,A)\subset\Gamma(x,A)\subset\Gamma^{o}(x,A^{\prime})$ for
$A<A^{\prime}$, we see

(7.14) $\Gamma(x)=\bigcup_{M\geq 0}\Gamma^{o}(x,M)$

and $(\Gamma^{o}(x,M))_{x\in E}$ will share most relevant properties with
$(\Gamma(x,M))_{x\in E}$, in particular, $(C,\delta_{\text{max}})$-convexity
and regularity.

##  8\. More on Convex Sets

In Section 9, we will establish a uniform choice of $\delta$ in the definition
of Glaeser stability which will allow us to choose polynomials from the
$\Gamma(x,A)$. The purpose of this section is to show that we may take
polynomials from the smaller $\Gamma(x,k,A)$ (where $k$ and $A$ will be chosen
appropriately).

######  Lemma 8.1.

Let $x_{0}\in E,A>0$, and ${\overline{k}}\in\mathbb{N}$. Suppose
$P_{0}\in\Gamma(x_{0},{\overline{k}},A)$ and
$P_{0}^{\prime}\in\Gamma(x_{0},A)$ such that
$\pi_{x_{0}}(P_{0}-P_{0}^{\prime})=0$. Then, there exists $C$, depending
solely on $m,n$ such that $P_{0}^{\prime}\in\Gamma(x_{0},{\overline{k}},CA)$.

###### Proof.

Let $x_{1},...,x_{{\overline{k}}}\in E$. By the fact that
$P_{0}\in\Gamma(x_{0},{\overline{k}},A)$, there exist $P_{j}\in\Gamma(x,A)$
($1\leq j\leq{\overline{k}}$) such that

(8.1) $|\partial^{\alpha}(P_{i}-P_{j})(x_{j})|\leq
A|x_{i}-x_{j}|^{m-|\alpha|}\text{ for }|\alpha|\leq m,0\leq
i,j,\leq{\overline{k}}.$

The conclusion will follow from establishing

(8.2) $|\partial^{\alpha}(P_{0}^{\prime}-P_{j})(x_{0})|\leq
A^{\prime}|x_{0}-x_{j}|^{m-|\alpha|}\text{ for }|\alpha|\leq m,1\leq
j\leq{\overline{k}}$

and

(8.3) $|\partial^{\alpha}(P_{0}^{\prime}-P_{j})(x_{j})|\leq
A^{\prime}|x_{0}-x_{j}|^{m-|\alpha|}\text{ for }|\alpha|\leq m,1\leq
j\leq{\overline{k}}.$

Since $P_{0},P_{0}^{\prime}\in\Gamma(x_{0},A)$, by (2.1) we have

(8.4)
$|\partial^{\alpha}P_{0}(x_{0})|,|\partial^{\alpha}P_{0}^{\prime}(x_{0})|\leq
A\text{ for }|\alpha|\leq m.$

Therefore, since $\partial^{\alpha}P_{0},\partial^{\alpha}P_{0}^{\prime}$ are
constant functions for $|\alpha|=m$, we have

(8.5) $|\partial^{\alpha}(P_{0}-P_{0}^{\prime})(y)|\leq
A_{1},y\in\mathbb{R}^{n},|\alpha|=m,$

where $A_{1}$ depends solely on $A,m,n$, and $\text{diam}E$. We are able to
ignore the dependence on $\text{diam}E$ in the future, as $E\subset Q_{0}$, a
cube of fixed length.

By (8.1) and (8.5), we see for $1\leq j\leq{\overline{k}}$ and $|\alpha|=m$

(8.6) $\displaystyle|\partial^{\alpha}(P_{0}^{\prime}-P_{j})(x_{0})|$
$\displaystyle\leq|\partial^{\alpha}(P_{0}^{\prime}-P_{0})(x_{0})|+|\partial^{\alpha}(P_{0}-P_{j})(x_{0})|$
(8.7) $\displaystyle\leq A_{1}+A|x_{0}-x_{j}|^{m-|\alpha|}$ (8.8)
$\displaystyle=(A_{1}+A)|x_{0}-x_{j}|^{m-|\alpha|},$

establishing (8.2) with $A^{\prime}=A_{1}+A$. (8.3) follows similarly.

It remains to check the cases of $|\alpha|\leq m-1$. Here, we see that (8.2)
follows immediately since $\pi_{x_{0}}(P_{0}-P_{0}^{\prime})=0$ implies
$\partial^{\alpha}P_{0}(x_{0})=\partial^{\alpha}P_{0}^{\prime}(x_{0})$ for
$|\alpha|\leq m-1$.

To establish (8.3), we observe that for $1\leq j\leq{\overline{k}}$ and
$|\alpha|\leq m-1$, by (8.1) and (8.4)

(8.9) $\displaystyle|\partial^{\alpha}(P_{0}^{\prime}-P_{j})(x_{j})|$
$\displaystyle\leq|\partial^{\alpha}(P_{0}^{\prime}-P_{0})(x_{j})|+|\partial^{\alpha}(P_{0}-P_{j})(x_{j})|$
(8.10) $\displaystyle\leq
A_{2}|x_{0}-x_{j}|^{m-|\alpha|}+A|x_{0}-x_{j}|^{m-|\alpha|},$

where $A_{2}$ depends solely on $A,m,n$. This gives (8.3) with
$A^{\prime}=A_{2}+A$.

∎

######  Lemma 8.2.

Let $A>0$ and $x\in E_{1}$. Suppose
$1+(D+1)\cdot{\tilde{k}}\leq{\overline{k}}$. If
$\overline{P}\in\text{int}\overline{\Gamma}(x,{\overline{k}},A)$, then there
exists $0<\delta<1$ such that if $x^{\prime}\in E_{1}\cap B(x,\delta)$, there
exists $\overline{P}^{\prime}\in\overline{\Gamma}(x^{\prime},{\tilde{k}},A)$
such that

(8.11)
$\left|\partial^{\alpha}(\overline{P}-\overline{P}^{\prime})(x)\right|,\left|\partial^{\alpha}(\overline{P}-\overline{P}^{\prime})(x^{\prime})\right|\leq
A\left|x-x^{\prime}\right|^{m-\left|\alpha\right|}\text{ for
}\left|\alpha\right|\leq m-1.$

###### Proof.

Choose
$\overline{P}_{0},...,\overline{P}_{\overline{d}}\in\overline{\Gamma}(x,{\overline{k}},A)$
such that $Conv(\overline{P}_{0},...,\overline{P}_{\overline{d}})$ is a
${\bar{d}}$-dimensional simplex containing
$B(\eta,\overline{P})\cap\pi_{x}W_{\Gamma}(x)$ for some $\eta>0$. Write
$\overline{P}_{j}=\pi_{x}P_{j}$ and $\overline{P}=\pi_{x}P$ for some
$P_{j},P\in\Gamma(x,{\overline{k}},A),0\leq j\leq{\bar{d}}.$

By definition of the $\Gamma(x,{\overline{k}},A)$, for any $x^{\prime}\in
E_{1}$, there exists $P_{j}^{\prime}\in\Gamma(x^{\prime},A)$ ($0\leq j\leq d$)
such that

(8.12) $|\partial^{\alpha}(P_{j}-P_{j}^{\prime})(x^{\prime})|\leq
A|x-x^{\prime}|^{m-|\alpha|}\text{ for }|\alpha|\leq m.$

In particular, since $|\partial^{\alpha}P_{j}(x)|\leq A$ for $|\alpha|\leq m$
by (2.1),

(8.13)
$|\partial^{\alpha}(\overline{P}_{j}-\overline{P}_{j}^{\prime})(x^{\prime})|\leq
A_{1}|x-x^{\prime}|^{m-|\alpha|}\text{ for }|\alpha|\leq m-1,$

where $\overline{P}_{j}^{\prime}=\pi_{x^{\prime}}P_{j}^{\prime}$ and $A_{1}$
is a constant depending solely on $A,m,n$.

Applying Lemma 6.3 to $\overline{P}_{0},...,\overline{P}_{\overline{d}}$, we
see that by taking $|x-x^{\prime}|$ sufficiently small, there exists
$0<\delta<1$ such that if $x^{\prime}\in E_{1}\cap B(x,\delta)$ and
$\overline{P}^{\prime}\in\pi_{x^{\prime}}W_{\Gamma}(x^{\prime})$ satisfies

(8.14) $|\partial^{\alpha}(\overline{P}-\overline{P}^{\prime})(x)|\leq
A_{1}|x-x^{\prime}|^{m-|\alpha|}\text{ for }|\alpha|\leq m-1,$

then
$\overline{P}^{\prime}\in\overline{\Lambda}(x^{\prime}):=Conv(\overline{P}_{0}^{\prime},...,\overline{P}_{d}^{\prime})\subset\overline{\Gamma}(x^{\prime},A)$.
$A_{1}$ will be chosen later and depend solely on $A,m,n$. Here, we implicitly
used Lemma 7.3 to show $\pi_{x^{\prime}}W_{\Gamma}$ was
${\bar{d}}$-dimensional, allowing us to apply Lemma 6.3.

Fix $x^{\prime}\in E_{1}\cap B(x,\delta)$. Given a finite set $S\subset E$,
define $S^{+}:=\left\\{x,x^{\prime}\right\\}\cup S$, and define

$\mathcal{K}(S):=\left\\{\overline{P}^{\prime}\in\overline{\Lambda}(x^{\prime}):\begin{matrix}\text{
There exists }P^{y}\in\Gamma(y,A)\text{ for all }y\in S^{+}\text{ such
that}\\\
P^{x}=P,\,\pi_{x^{\prime}}P^{x^{\prime}}=\overline{P}^{\prime},\,\text{ and
}\\\ \left|\partial^{\alpha}(P^{y}-P^{z})(z)\right|\leq
A\left|y-z\right|^{{m-\left|\alpha\right|}}\text{ for }\left|\alpha\right|\leq
m,\,y,z\in S^{+}.\end{matrix}\right\\}$

Each $\mathcal{K}(S)$ is compact, convex subset of $\overline{\mathcal{P}}$,
which has dimension $\leq D$. Given $S\subset S^{\prime}$, we have
$\mathcal{K}(S^{\prime})\subset\mathcal{K}(S)$. Given $S^{+}\subset E$ with
$\\#(S^{+})\leq{\overline{k}}+1$, we see from $P\in\Gamma(x,{\overline{k}},A)$
there exists $\tilde{P}\in\mathcal{P}$ such that $\pi_{x^{\prime}}\tilde{P}$
satisfies the conditions for membership in $\mathcal{K}(S^{+})$, except it is
only known so far that $\pi_{x^{\prime}}\tilde{P}\in\overline{\mathcal{P}}$,
rather than $\pi_{x^{\prime}}\tilde{P}\in\overline{\Lambda}(x^{\prime})$.
However, $|\partial^{\alpha}(P-P^{x^{\prime}})(x^{\prime})|\leq
A|x-x^{\prime}|^{m-|\alpha|}$ implies (8.14) with $A_{1}$ chosen accordingly;
thus, it is required that
$\pi_{x^{\prime}}\tilde{P}\in\overline{\Lambda}(x^{\prime})$ thus
$\pi_{x^{\prime}}\tilde{P}\in\mathcal{K}(S^{+})$ and $\mathcal{K}(S^{+})$ is
nonempty.

Suppose $S_{1},\cdots,S_{D+1}\subset E$ with $\\#(S_{i})\leq\tilde{k}$ for
$i=1,\cdots,D+1$. Then $S:=S_{1}\cup\cdots\cup S_{D+1}$ satisfies
$\\#(S)\leq{\overline{k}}+1$. Hence, $\mathcal{K}(S)\neq\varnothing$, and
$\mathcal{K}(S)\subset\mathcal{K}(S_{i})$ for each $i$. Thus,
$\bigcap_{i=1}^{D+1}\mathcal{K}(S_{i})\neq\varnothing$. By Helly’s theorem,
there exists $\overline{P}^{\prime}\in\mathcal{K}(S)$ for every $S\subset E$
with $\\#(S)\leq\tilde{k}$. It follows that
$\overline{P}^{\prime}\in\overline{\Gamma}(x^{\prime},\tilde{k},A)$. Taking
$S=\varnothing$ and using the fact that
$\overline{P}^{\prime}\in\mathcal{K}(S)$, we see that the first conclusion
follows.

∎

######  Lemma 8.3.

Suppose $A>0$ and $1+(D+1)\tilde{k}\leq{\overline{k}}$. Given $x\in E_{1}$,
and $\overline{Q}\in\text{int}\overline{\Gamma}(x,{\overline{k}},A)$, there
exist $\epsilon_{0},\delta_{0}>0$ such that for any $x\in E_{1}\cap
B(x,\delta_{0})$ and any
$\overline{Q}^{\prime}\in\overline{\Gamma}(x^{\prime},A)$, if
$|\partial^{\alpha}(\overline{Q}-\overline{Q}^{\prime})(x)|\leq\epsilon_{0}$
for $|\alpha|\leq m-1$, then
$\overline{Q}^{\prime}\in\overline{\Gamma}(x^{\prime},\tilde{k},A)$.

###### Proof.

Choose
$\overline{Q}_{0},...,\overline{Q}_{{\bar{d}}}\in\text{int}\overline{\Gamma}(x,{\overline{k}},A)$
which form a ${\bar{d}}$-dimensional simplex with $\overline{Q}$ lying in its
relative interior. Let $\eta>0$ be as in Lemma 6.3.

Applying Lemma 8.2 to each of the $\overline{Q}_{j}$ and taking $\delta_{0}$
to be the minimum of the associated $\delta$’s, we see that for any
$x^{\prime}\in E_{1}\cap B(x,\delta_{0})$, there exist
$\overline{Q}_{0}^{\prime},...,\overline{Q}_{{\bar{d}}}^{\prime}\in\overline{\mathcal{P}}$
such that

(8.15)
$\overline{Q}_{j}^{\prime}\in\overline{\Gamma}(x^{\prime},{\overline{k}},A)$

and

(8.16)
$\left|\partial^{\alpha}(\overline{Q}_{j}-\overline{Q}_{j}^{\prime})(x)\right|,\left|\partial^{\alpha}(\overline{Q}_{j}-\overline{Q}_{j}^{\prime})(x^{\prime})\right|\leq
A\left|x-x^{\prime}\right|^{m-\left|\alpha\right|}\text{ for
}\left|\alpha\right|\leq m-1.$

Taking $\delta_{0}>0$ small enough, by (8.16), Lemma 6.3 applies. (By Lemma
7.3 and the definition of strata,
$\dim\overline{\Gamma}(x^{\prime},A)=\dim\overline{\Gamma}(x,A)={\bar{d}}$.)
Thus, there exists $\epsilon_{0}>0$ such that if
$\overline{Q}^{\prime}\in\overline{\Gamma}(x^{\prime},A)$, if
$|\partial^{\alpha}(\overline{Q}-\overline{Q}^{\prime})(x)|\leq\epsilon_{0}$
for $|\alpha|\leq m-1$, then $\overline{Q}^{\prime}\in
Conv(\overline{Q}_{0}^{\prime},...,\overline{Q}_{{\bar{d}}}^{\prime})\subset\overline{\Gamma}(x^{\prime},\tilde{k},A)$.
∎

######  Lemma 8.4.

Suppose $A,A_{1}>0$ and $1+(D+1)\tilde{k}\leq{\overline{k}}$. Given $x\in
E_{1}$, and $Q\in\text{int}\Gamma(x,{\overline{k}},A)$, there exist
$\epsilon,\delta>0$ such that for any $x^{\prime}\in E_{1}\cap B(x,\delta)$
and any $Q^{\prime}\in\Gamma(x^{\prime},A)$, if

(8.17) $|\partial^{\alpha}(Q-Q^{\prime})(x)|\leq\epsilon\text{ for
}|\alpha|\leq m-1,$

and

(8.18) $|\partial^{\alpha}Q^{\prime}(x)|\leq A\text{ for }|\alpha|=m,$

then $Q^{\prime}\in\Gamma(x^{\prime},\tilde{k},CA)$, with $C$ depending only
on $m,n$.

###### Proof.

Let $x\in E_{1}$. Let $\epsilon_{0},\delta_{0}$ be as in Lemma 8.3 with
$A=A_{1}$. Let $\epsilon$ and $\delta$ be small numbers to be determined,
depending only on the parameters appeared above.

Fix $Q\in\Gamma(x,{\overline{k}},A_{1})$, $x^{\prime}\in E_{1}\cap
B(x,\delta)$, and $Q^{\prime}\in\Gamma(x^{\prime},A)$ satisfying (8.17) and
(8.18).

Since $Q\in\Gamma(x,{\overline{k}},A_{1})$, we have
$\left|\partial^{\alpha}Q(x)\right|\leq A_{1}$ for $\left|\alpha\right|\leq
m$. Therefore, (8.17) and (8.18) imply that
$\left|\partial^{\alpha}Q^{\prime}(x)\right|\leq A_{3}$ for
$\left|\alpha\right|\leq m$. Taking $\delta\leq 1$, we have

(8.19) $\left|\partial^{\alpha}Q^{\prime}(x^{\prime})\right|\leq A_{4}\text{
for }{\left|\alpha\right|\leq m}.$

We set $\overline{Q}:=\pi_{x}Q$ and
$\overline{Q}^{\prime}:=\pi_{x^{\prime}}Q^{\prime}$, immediately seeing that
by taking $\delta\leq\delta_{0}$, we have

(8.20)
$\overline{Q}\in\mathrm{int}\,\overline{\Gamma}(x,{\overline{k}},A_{1}),\,x^{\prime}\in
E_{1}\cap B(x,\delta_{0}),\text{ and
}\overline{Q}^{\prime}\in\overline{\Gamma}(x^{\prime},A).$

By Taylor’s theorem and (8.19), we have

(8.21)
$\left|\partial^{\alpha}(\overline{Q}^{\prime}-Q^{\prime})(x)\right|\leq
A_{3}\left|x-x^{\prime}\right|^{m-\left|\alpha\right|}\leq
A_{3}\delta^{m-\left|\alpha\right|}\leq A_{3}\delta\text{
for}{\left|\alpha\right|\leq m}.$

Taking $\epsilon$ and $\delta$ to be sufficiently small, we see from (8.17)
and (8.21) that

(8.22)
$\left|\partial^{\alpha}(\overline{Q}^{\prime}-\overline{Q})(x)\right|\leq\epsilon_{0}\text{
for }\left|\alpha\right|\leq m-1.$

In view of (8.20) and (8.22), we see that Lemma 8.3 applies. Hence,
$\overline{Q}^{\prime}\in\overline{\Gamma}(x^{\prime},\tilde{k},A_{4})$. This
means that there exists some $\tilde{Q}\in\Gamma(x^{\prime},\tilde{k},A_{4})$
such that $\overline{Q}^{\prime}=\pi_{x^{\prime}}\tilde{Q}$. Now, since
$\pi_{x^{\prime}}(\tilde{Q}-Q^{\prime})\equiv 0$, we see from Lemma 8.1 that
$Q^{\prime}\in\Gamma(x^{\prime},\tilde{k},A_{5})$. ∎

##  9\. Uniform delta and modulus of continuity

The purpose of this section is to show that, given $\epsilon>0$, we may take
the $\delta$ in the definition of Glaeser stable to be uniform in $x$ and
$P_{0}$ (with some change in ${k^{\sharp}}$), then use this fact and the
results of Section 8 to construct a modulus of continuity that will let us
apply Theorem 5.2.

Given $(x,P),(x^{\prime},P^{\prime})\in E\times\mathcal{P}$, we let

(9.1)
$\text{dist}\left((x,P),(x^{\prime},P^{\prime})\right):=|x-x^{\prime}|+\max_{\alpha}|\partial^{\alpha}(P-P^{\prime})(x)|+\max_{\alpha}|\partial^{\alpha}(P-P^{\prime})(x^{\prime})|.$

Note that the above turns $E\times\mathcal{P}$ into a metric space. While the
above description of $\text{dist}(\cdot,\cdot)$ will be most useful, the
metric defined by

(9.2)
$\rho((x,P),(x^{\prime},P^{\prime})):=|x-x^{\prime}|+\max_{\alpha}|\partial^{\alpha}(P-P^{\prime})(0)|.$

will be useful for establishing a crucial property of
$\text{dist}(\cdot,\cdot)$.

######  Lemma 9.1.

The metrics $\text{dist}(\cdot,\cdot)$ and $\rho(\cdot,\cdot)$ are equivalent,
that is, given compact $E\subset\mathbb{R}^{n}$, there exists $C>0$ such that

(9.3)
$C^{-1}\rho((x,P),(x^{\prime},P^{\prime}))\leq\text{dist}((x,P),(x^{\prime},P^{\prime}))\leq
C\rho((x,P),(x^{\prime},P^{\prime}))$

whenever $x,x^{\prime}\in E,P,P^{\prime}\in\mathcal{P}$.

In particular, a set is compact in the dist metric if and only if it is closed
and bounded in the dist metric.

###### Proof.

Write

(9.4) $P-P^{\prime}(y)=\sum_{\beta}c_{\beta}x^{\beta},$

so that

(9.5)
$\rho((x,P),(x^{\prime},P^{\prime}))=|x-x^{\prime}|+\max_{\alpha}|c_{\alpha}|$

and

(9.6) $\partial^{\alpha}(P-P^{\prime})(y)=\sum_{\begin{subarray}{c}|\beta|\leq
m\\\ \beta\geq\alpha\end{subarray}}c_{\beta}\beta!y^{\beta-\alpha}.$

Then, letting $M=\sup_{y\in E}|y|<\infty$,

$\displaystyle\text{dist}((x,P),(x^{\prime},P^{\prime}))$
$\displaystyle=|x-x^{\prime}|+\max_{\alpha}|\partial^{\alpha}(P-P^{\prime})(x)|+\max_{\alpha}|\partial^{\alpha}(P-P^{\prime})(x^{\prime})|$
$\displaystyle=|x-x^{\prime}|+\max_{\alpha}\left|\sum_{\begin{subarray}{c}|\beta|\leq
m\\\
\beta\geq\alpha\end{subarray}}c_{\beta}\beta!x^{\beta-\alpha}\right|+\max_{\alpha}\left|\sum_{\begin{subarray}{c}|\beta|\leq
m\\\
\beta\geq\alpha\end{subarray}}c_{\beta}\beta!(x^{\prime})^{\beta-\alpha}\right|$
$\displaystyle\leq|x-x^{\prime}|+2DM^{m}\max_{|\beta|\leq
m}\beta!\max_{|\beta|\leq m}|c_{\beta}|$
$\displaystyle\leq\left[2DM^{m}\max_{|\beta|\leq
m}\beta!\right]\rho((x,P),(x^{\prime},P^{\prime})).$

The reverse direction may be proven analogously.

The metric $\rho(\cdot,\cdot)$ is by definition equivalent to a Euclidean
metric on $E\times\mathcal{P}$ identified in the natural way with
$E\times\mathbb{R}^{D}$. By the above, so is the metric
$\text{dist}(\cdot,\cdot)$; thus, the Heine-Borel theorem applies, which is
the form of our conclusion. ∎

For Lemmas 9.2 through 9.6, fix a Glaeser stable, convex bundle $(H(x))_{x\in
E}$ with lowest stratum $E_{1}$.

######  Lemma 9.2.

Suppose $1+(D+1)\overline{k}\leq k^{\sharp}$. Let $x\in E,$
$P\in\mathrm{int}H(x)$, and $\epsilon>0$ be given. Then there exists
$0<\delta<1$ (depending on $P$) such that for every $x^{\prime}\in E_{1}\cap
B(x,\delta)$, there exists $P^{\prime}\in H^{\prime}(x^{\prime})$ such that

(9.7)
$|\partial^{\alpha}(P-P^{\prime})(x)|\leq\epsilon|x-x^{\prime}|^{m-|\alpha|}$

and given $x_{j}^{\prime}\in E\cap B(x,\delta)$ ($1\leq j\leq\overline{k}$),
there exist $P_{j}^{\prime}\in H(x_{j})$ such that

(9.8)
$|\partial^{\alpha}(P_{i}^{\prime}-P_{j}^{\prime})(x_{j}^{\prime})|\leq\epsilon|x_{i}^{\prime}-x_{j}^{\prime}|^{m-|\alpha|},0\leq
i,j,\leq{\overline{k}}.$

We mimic the proof of Lemma 6.1 in [6], the most significant change being the
introduction of simplices so the compactness hypothesis of Helly’s theorem is
satisfied.

###### Proof.

Recall that the bundle $(H(x))_{x\in E}$ is assumed to be Glaeser stable.

Suppose $x^{\prime}=x$, then we may take $P^{\prime}\equiv P$ and (9.7)
follows. (9.8) follows from Glaeser stability.

Suppose $x^{\prime}\neq x$. Since $P\in\text{int}H(x)$, choose
$P_{0},...,P_{d}\in H(x)$ which form a $d$-dimensional simplex containing
$B(P,\delta)\cap H(x)$. Let $\eta$ be as in the conclusion of Lemma 6.3,
applied to $P_{0},...,P_{d}$.

Let $0<\epsilon^{\prime}<\min\\{\epsilon,\eta\\}$ and by the Glaeser stability
of $(H(x))_{x\in E}$ pick $0<\delta<\eta$ such that for $j=0,...,d$, if
$x^{\prime}\in E\cap B(x^{\prime},\delta)$ there exists $P_{j}^{\prime}\in
H(x^{\prime})$ such that

(9.9)
$|\partial^{\alpha}(P_{j}-P_{j}^{\prime})(x)|,|\partial^{\alpha}(P_{j}-P_{j}^{\prime})(x^{\prime})|\leq\epsilon^{\prime}|x-x^{\prime}|^{m-|\alpha|}\leq\eta\text{
for }|\alpha|\leq m.$

Thus, if $P^{\prime}\in H(x^{\prime})$ such that

(9.10)
$|\partial^{\alpha}(P-P^{\prime})(x)|,|\partial^{\alpha}(P-P^{\prime})(x^{\prime})|\leq\epsilon|x-x^{\prime}|^{m-|\alpha|}\leq\eta\text{
for }|\alpha|\leq m,$

then
$P^{\prime}\in\Lambda(x^{\prime}):=Conv(P_{0}^{\prime},...,P_{j}^{\prime})$
(as the definition of lowest stratum implies $\dim H(x^{\prime})=\dim H(x)$).

For any finite set $S\subset E\cap B(x,\delta)$ with
$S\supset\left\\{x,x^{\prime}\right\\}$, define

$\mathcal{K}(S):=\left\\{P^{\prime}\in\Lambda(x^{\prime}):\begin{matrix}\text{
There exists a map }y\mapsto P^{y}\text{ from }S\text{ to }\mathcal{P}\text{
such that}\\\ P^{x}=P,\,P^{x^{\prime}}=P^{\prime},\,P^{y}\in H(y)\text{ for
}y\in S,\text{ and }\\\
\left|\partial^{\alpha}(P^{y}-P^{z})(z)\right|\leq\epsilon^{\prime}\left|y-z\right|^{m-\left|\alpha\right|}\text{
for }\left|\alpha\right|\leq m,\,y,z\in S.\end{matrix}\right\\}$

Each $\mathcal{K}(S)$ is a compact, convex subset of $\mathcal{P}$, which has
dimension $D$.

Suppose we are given $S_{1},\cdots,S_{D+1}\subset E\cap B(x,\delta)$ , each
containing $x$ and $x^{\prime}$, with $\\#(S_{i})\leq{\overline{k}}+2$ for
each $i$. Then $S:=\bigcup_{i=1}^{D}S_{i}\subset E\cap B(x,\delta)$, with
$x,x^{\prime}\in S$, and $\\#(S)\leq 2+(D+1){\overline{k}}\leq
1+{k^{\sharp}}$. Therefore, by the Glaeser stability of $(H(x))_{x\in E}$,
there exists $y\mapsto P^{y}$, with $P^{x}=P$, $P^{y}\in H(y)$ for each $y\in
S$, and

(9.11)
$\left|\partial^{\alpha}(P^{y}-P^{z})(z)\right|\leq\epsilon^{\prime}\left|y-z\right|^{{m-\left|\alpha\right|}}\text{
for }\left|\alpha\right|\leq m,y,z\in S.$

(9.10) forces $P^{x^{\prime}}\in\Lambda(x^{\prime})$.

It is clear that $P^{x^{\prime}}\in\mathcal{K}(S_{i})$ for each
$i=1,\cdots,D+1$. Therefore, $\mathcal{K}(S_{1}),\cdots,\mathcal{K}(S_{D+1})$
have nonempty intersection. By Helly’s theorem, there exists
$P^{\prime}\in\mathcal{K}(S^{\prime})\subset H(x^{\prime})$, whenever
$S^{\prime}\subset E\cap B(x,\delta)$, with $x,x^{\prime}\in S^{\prime}$ and
$\\#(S^{\prime})\leq{\overline{k}}+2$. (9.7) and (9.8) then follow from the
definition of $\mathcal{K}(S^{\prime})$. ∎

######  Definition 9.1.

Let $(H(x))_{x\in E}$ be a Glaeser stable bundle. Given $x_{0}\in E$,
$P_{0}\in H(x_{0})$, $\epsilon>0$ and ${\overline{k}}\in\mathbb{N}$, we define
$\Delta(x_{0},P_{0};\epsilon,{\overline{k}})$ to be the supremum over all
$\delta>0$ such that the following holds:

If $x_{1},...,x_{k}\in E\cap B(x_{0},\delta)$, there exist
$P_{1},...,P_{k}\in\mathcal{P}$ such that

(9.12) $P_{j}\in H(x_{j}),1\leq j\leq{\overline{k}}$

and

(9.13)
$|\partial^{\alpha}(P_{i}-P_{j})(x_{j})|\leq\epsilon|x_{i}-x_{j}|^{m-|\alpha|}\text{
for }|\alpha|\leq m,0\leq i,j\leq{\overline{k}}.$

Note that since the $x_{i}$ are taken to be in the open ball of radius
$\delta$, the above holds with
$\delta=\Delta(x_{0},P_{0};\epsilon,{\overline{k}})$.

######  Lemma 9.3.

Let $x_{0}\in E$ and ${\overline{k}}\leq{k^{\sharp}}$. Suppose
$P_{0}^{(0)},...,P_{0}^{(L)}\in H(x)$. Let
$P_{0}=\sum_{l=0}^{L}\lambda_{l}P_{0}^{(l)}$ for $\lambda_{l}\in[0,1]$. Then,
$\Delta(x,P_{0};{\overline{k}},\epsilon)\geq\min_{0\leq l\leq
L}\Delta(x,P_{0}^{(l)};{\overline{k}},\epsilon)$.

###### Proof.

Let $\epsilon>0$. Set $\delta_{0}:=\min_{0\leq l\leq
L}\Delta(x,P_{0}^{(l)};{\overline{k}},\epsilon)$.

Let $x_{1},...,x_{\overline{k}}\in E\cap B(x,\delta_{0})$. By the definition
of Glaeser stability, for each $0\leq l\leq L$ there exist
$P_{1}^{(l)},...,P_{{\overline{k}}}^{(l)}\in H(x_{j})$, and

(9.14)
$\left|\partial^{\alpha}(P_{i}^{(l)}-P_{j}^{(l)})(x_{j})\right|\leq\epsilon\left|x_{i}-x_{j}\right|^{m-\left|\alpha\right|}\text{
for }\left|\alpha\right|\leq m,\,0\leq i,j\leq{\overline{k}},0\leq l\leq L.$

For $1\leq j\leq{\overline{k}}$ and $0\leq l\leq L$, we set

$P_{j}:=\sum_{l=0}^{L}\lambda_{l}P_{j}^{(l)}.$

Since $H(x_{j})$ is convex for each $j$, we have $P_{j}\in H(x_{j})$.
Furthermore, thanks to (9.14), we have

$\begin{split}\left|\partial^{\alpha}(P_{i}-P_{j})(x_{j})\right|&\leq\sum_{0\leq
l\leq
L}\lambda_{j}|\partial^{\alpha}(P_{i}^{(l)}-P_{j}^{(l)})(x_{j})|\leq\epsilon\left|x_{i}-x_{j}\right|^{m-\left|\alpha\right|}\end{split}$

for $\left|\alpha\right|\leq m$, $0\leq i,j\leq{\overline{k}}$.

∎

######  Lemma 9.4.

Suppose $1+(D+1)\overline{k}\leq k^{\sharp}$. Let $x\in E_{1}$ and
$P\in\mathrm{int}H(x)$. There exists $\epsilon_{0}>0$ such that when
$0<\epsilon<\epsilon_{0}$, there exist $\delta,\eta>0$ such that whenever
$x^{\prime}\in E_{1}$, $P^{\prime}\in W_{H}(x^{\prime})$ and

(9.15) $\text{dist}((x,P),(x,P^{\prime}))<\eta,$

we have $P^{\prime}\in H(x^{\prime})$ and

(9.16) $\Delta(x^{\prime},P^{\prime};{\overline{k}},\epsilon)\geq\delta.$

###### Proof.

Note that if $d=0$, then $H(x)$ consists of a single polynomial for any $x\in
E_{1}$ and the conclusion follows from Lemma 9.2.

Suppose $d\geq 1$ and let $P_{0},...,P_{d}\in\text{int}H(x)$ which form a
nondegenerate affine simplex with $P$ in its relative interior. Let
$\eta_{0}>0$ be as in the conclusion of Lemma 6.3. Noting that $\eta_{0}$ is
dependent solely on $P$, set $\epsilon_{0}=\eta_{0}$.

For each $P_{j}$, let $\delta_{j}>0$ be as in the conclusion of Lemma 9.2 and
take $\delta=\frac{1}{2}\min\\{\delta_{0},...,\delta_{d},1\\}$. Now take
$\eta=\min\\{\delta,\eta_{0}\\}$.

Now, suppose $x^{\prime}\in E_{1}\cap B(x,\delta)$, $P^{\prime}\in
H(x^{\prime})$ and $\text{dist}((x,P),(x^{\prime},P^{\prime}))<\eta$. Then,

(9.17) $|x-x^{\prime}|\leq\delta$

and

(9.18)
$|\partial^{\alpha}(P-P^{\prime})(x)|,|\partial^{\alpha}(P-P^{\prime})(x^{\prime})|<\eta\text{
for }|\alpha|\leq m$

By (9.17) and Lemma 9.2, there exist $P_{0}^{\prime},...,P_{d}^{\prime}\in
H(x^{\prime})$ such that

(9.19)
$|\partial^{\alpha}(P_{j}-P_{j}^{\prime})(x_{0})|,|\partial^{\alpha}(P_{j}-P_{j}^{\prime})(x^{\prime})|\leq\epsilon|x^{\prime}-x_{0}|^{m-|\alpha|}\leq\eta_{0}\text{
for }|\alpha|\leq m$

and

(9.20)
$\Delta(x^{\prime},P_{j}^{\prime};{\overline{k}},\epsilon)\geq\delta,0\leq
j\leq d.$

By Lemma 6.3, the definition of lowest stratum, and (9.18), $P^{\prime}$ lies
in the convex hull of $P_{0}^{\prime},...,P_{d}^{\prime}$. Thus, by Lemma 9.3,

(9.21) $\Delta(x^{\prime},P^{\prime};{\overline{k}},\epsilon)\geq\delta.$

∎

######  Lemma 9.5.

Let $x\in E_{1}$ and $P\in\mathrm{int}H(x)$. If $\epsilon>0$, then there
exists $\tilde{\eta}>0$ such that whenever $x^{\prime}\in E_{1}\cap
B(x,\tilde{\eta})$, there exists $P^{\prime}\in H(x^{\prime})$ such that

(9.22) $\text{dist}((x,P),(x^{\prime},P^{\prime}))<\epsilon.$

###### Proof.

By the definition of Gleaser stability, there exists
$0<\tilde{\eta}<\min\\{1,\epsilon/2\\}$ such that whenever $x^{\prime}\in
E_{1}\cap B(x,\tilde{\eta})$, there exists $P^{\prime}\in H(x^{\prime})$ such
that

(9.23)
$|\partial^{\alpha}(P-P^{\prime})(x)|,|\partial^{\alpha}(P-P^{\prime})(x^{\prime})|\leq\epsilon/2|x-x^{\prime}|^{m-|\alpha|}\leq\epsilon/2\text{
for }|\alpha|\leq m.$

Combining (9.23) with the fact $|x-x^{\prime}|<\tilde{\eta}<\epsilon/2$ we
obtain the desired conclusion. ∎

######  Lemma 9.6.

Suppose $1+(D+1)\overline{k}\leq k^{\sharp}$. Let $\epsilon>0$ and $K_{0}$ be
a compact subset of $\mathbb{R}^{n}\times\mathcal{P}$ such that whenever
$(x,P)\in K_{0}$, $P\in\mathrm{int}H(x)$. Then, there exists $\delta>0$ such
that whenever $(x_{0},P_{0})\in K_{0}$ and $x_{1},...,x_{k}\in E_{1}\cap
B(x_{0},\delta)$, there exists $P_{1},...,P_{{\overline{k}}}\in H(x)$ such
that

(9.24)
$|\partial^{\alpha}(P_{i}-P_{j})(x_{j})|\leq\epsilon|x_{i}-x_{j}|^{m-|\alpha|}\text{
for }|\alpha|\leq m,0\leq i,j,\leq{\overline{k}}.$

###### Proof.

Let $(x,P)\in K_{0}$. Then, since $P$ lies in the relative interior of $H(x)$,
we may apply Lemma 9.4 to obtain $\eta_{x},\delta_{x},(\epsilon_{0})_{x}>0$
such that whenever $(x^{\prime},P^{\prime})$ lies in

(9.25) $B_{K_{0}}((x,P),\eta_{x}):\\{(x^{\prime},P^{\prime})\in
K_{0}:\text{dist}((x,P),(x^{\prime},P^{\prime}))<\eta_{x}\\},$

and $0<\epsilon<(\epsilon_{0})_{x}$ we have

(9.26) $\Delta(x^{\prime},P^{\prime};{\overline{k}},\epsilon)\geq\delta_{x}.$

Thus, $\\{B_{K_{0}}((x,P),\eta_{x})\\}_{(x,P)\in K}$ forms an open cover of
$K_{0}$. By compactness of $K_{0}$, we obtain a finite subcover
$\\{B_{K_{0}}((x_{1},P_{1}),\eta_{x_{1}}),...,B_{K_{0}}((x_{l},P_{l}),\eta_{x_{l}})\\}$.
Taking $\delta=\min\\{\delta_{x_{1}},...,\delta_{x_{l}}\\}>0$ and requiring
$0<\epsilon<\min\\{(\epsilon_{0})_{x_{1}},...,(\epsilon_{0})_{x_{l}}\\}$, we
see that

(9.27) $\Delta(x^{\prime},P^{\prime};{\overline{k}},\epsilon)\geq\delta$

for all $(x^{\prime},P^{\prime})\in K_{0}$, which is equivalent to our desired
conclusion.

For $\epsilon\geq\min\\{(\epsilon_{0})_{x_{1}},...,(\epsilon_{0})_{x_{l}}\\},$
apply the conclusion for
$\frac{1}{2}\min\\{(\epsilon_{0})_{x_{1}},...,(\epsilon_{0})_{x_{l}}\\}$ and
observe

(9.28)
$\Delta(x^{\prime},P^{\prime};{\overline{k}},\epsilon)\geq\Delta\left(x^{\prime},P^{\prime};{\overline{k}},\frac{1}{2}\min\\{(\epsilon_{0})_{x_{1}},...,(\epsilon_{0})_{x_{l}}\\}\right)\geq\delta.$

∎

We now consider the case where $(H(x))_{x\in E}$ is the shape field
$(\Gamma(x,M))_{x\in E}$. Let $x\in E_{1}$ and
$P\in\text{int}\Gamma(x,{k^{\sharp}},2A^{\sharp})$.

Let $\epsilon_{0}=(\epsilon_{0})_{(x,P)}$ be as in Lemma 9.4 applied to the
Glaeser stable bundle $(\Gamma(x,2A^{\sharp}))_{x\in E}$. (See Lemma 7.5.) Let
$\eta_{(x,P)}$ and $\delta_{(x,P)}$ be the corresponding $\eta$ and $\delta$
from the conclusion of Lemma 9.4 with $\epsilon$ taken as
$(\epsilon_{0})_{(x,P)}/2$.

By Lemma 9.5, there exists $0<\tilde{\eta}_{(x,P)}<\eta_{(x,P)}$ such that
whenever $x^{\prime}\in E_{1}\cap B(x,\tilde{\eta})$, there exists
$P^{\prime}\in\Gamma(x^{\prime},A^{\sharp})$ such that

(9.29) $\text{dist}((x,P),(x^{\prime},P^{\prime}))<\eta_{(x,P)}/4.$

Furthermore, by Lemma 8.4, we may take $\eta_{(x,P)}$ and
$\tilde{\eta}_{(x,P)}$ small enough that if $x^{\prime}\in E_{1}$ and
$P\in\Gamma(x^{\prime})$ satisfies

(9.30) $\text{dist}((x,P),(x^{\prime},P^{\prime}))\leq\eta_{(x,P)},$

then $P^{\prime}\in\Gamma(x^{\prime},\tilde{k},CA^{\sharp})$, where $C$ is as
in the conclusion of Lemma 8.4 and $\tilde{k}$ satisfying

(9.31) $1+(D+1)\tilde{k}\leq{k^{\sharp}}.$

Consider the collection of balls
$B(x,\tilde{\eta}_{(x,P)}/4)\subset\mathbb{R}^{n}$, with $x$ ranging over
$E_{1}$ and given $x$, $P$ ranging over
$\text{int}\Gamma(x,{k^{\sharp}},2A^{\sharp})$. This forms an open cover of
$E_{1}$. $E_{1}$ is compact, so consider a finite subcover

(9.32)
$\left\\{B(x_{1},\tilde{\eta}_{(x_{1},P_{1})}/4),...,B(x_{J},\tilde{\eta}_{(x_{J},P_{J})}/4)\right\\}.$

Define

(9.33)
$K_{j}=\\{(x,P)\in\overline{B}((x_{j},P_{j}),{\eta}_{(x_{j},P_{j})}/2):x\in
E_{1}\cap\overline{B}(x,\tilde{\eta}/2),P\in\Gamma(x,2A^{\sharp})\\},$

where $\overline{B}$ is used to refer to the closed ball, and

(9.34) $K=\bigcup_{j=1}^{J}K_{j}\subset E_{1}\times\mathcal{P}.$

Note that Lemma 8.4 implies that

(9.35) $\text{if }(x,P)\in K,\text{ then
}P\in\Gamma(x,\tilde{k},CA^{\sharp}).$

with $C$ depending only on $m,n$.

######  Lemma 9.7.

$K$ is compact under the topology induced by the dist metric defined in (9.1).

###### Proof.

It suffices to show $K_{j}$ is compact for arbitrary $1\leq j\leq J$. Let
$1\leq j_{0}\leq J$. By Lemma 9.1, it suffices to show $K_{j_{0}}$ is closed
and bounded; boundedness follows immediately from (2.1).

To show $K_{j_{0}}$ is closed, let $x\in\mathbb{R}^{n},P\in\mathcal{P}$, and
$(x_{1},P_{1}),(x_{2},P_{2}),...\in K_{j_{0}}$ such that

(9.36) $|x_{i}-x|,\text{dist}((x_{i},P_{i}),(x,P))\to 0\text{ as }i\to\infty.$

It follows that
$(x,P)\in\overline{B}((x_{j_{0}},P_{j_{0}}),{\eta}_{(x_{j_{0}},P_{j_{0}})}/2)$
and $x\in\overline{B}(x_{j_{0}},\tilde{\eta}_{(x_{j_{0}},P_{j_{0}})}/2)$ since
the same is true for each $(x_{i},P_{i})$.

By compactness of $E_{1}$ established by Lemma 7.4, we have $x\in E_{1}$. Our
goal is to show $P\in\Gamma(x,2A^{\sharp})$. By Lemma 9.4 and the definition
of $K$, to show $P\in\Gamma(x,2A^{\sharp})$, it suffices to show $P\in
W_{\Gamma}(x)$.

By the compactness of the Grassmanian of $d$-dimensional hyperplanes of
$\mathcal{P}$ having nontrivial intersection with
$\\{|\partial^{\alpha}P(y)|\leq 2A^{\sharp}\text{ for }|\alpha|\leq m\\}$ for
some $y\in E$, we may suppose by passing to a subsequence that
$W_{\Gamma}(x_{i})$ converges to some $d$-dimensional subspace
$W\subset\mathcal{P}$.

If $W=W(x)$ we are done, so suppose for the sake of contradiction that $W\neq
W_{\Gamma}(x)$. Then there exists $\tilde{P}\in\Gamma(x,2A^{\sharp})$ such
that $\tilde{P}\notin W_{\Gamma}(x)$, hence there cannot exist a sequence
$P_{i}\in W_{\Gamma}(x_{i})$ satisfying $P_{i}\to\tilde{P}$. This contradicts
the Glaeser-stable property of $(\Gamma(x,2A^{\sharp}))_{x\in E}$ established
in Lemma 7.5. Thus, $W=W_{\Gamma}(x)$. ∎

######  Lemma 9.8.

Let $\tilde{k}$ be as in (9.31). There exists $\gamma>0$ such that:

For all $x\in E_{1}$, there exists $P\in\Gamma(x,\tilde{k},CA^{\sharp})$ such
that $(x,P)\in K$ and if $x^{\prime}\in
E_{1},P^{\prime}\in\Gamma(x^{\prime},2A^{\sharp})$ such that
$\text{dist}((x,P),(x^{\prime},P^{\prime}))<\gamma$, then
$(x^{\prime},P^{\prime})\in K$.

###### Proof.

Returning to the notation of the construction of $K$, choose $1\leq j\leq J$
such that $x\in B(x_{j},\tilde{\eta}_{(x_{j},P_{j})}/4)$. By Lemma 9.5 and the
construction of $K$, there exists $P\in\Gamma(x,2A^{\sharp})$ such that

(9.37) $\text{dist}((x_{j},P_{j}),(x,P))<\eta_{(x_{j},P_{j})}/4,$

thus $(x,P)\in K$ and by (9.35), $P\in\Gamma(x,\tilde{k},CA^{\sharp})$.

Now suppose $x^{\prime}\in E_{1}\cap
B(x,\tilde{\eta}_{(x_{j},P_{j})}/4),P^{\prime}\in\Gamma(x^{\prime})$ such that
$\text{dist}((x,P),(x^{\prime},P^{\prime}))<{\eta}_{(x_{j},P_{j})}/2$. Then by
the triangle inequality, we have

(9.38) $|x_{j}-x^{\prime}|<\tilde{\eta}_{(x_{j},P_{j})}$

and

(9.39)
$\text{dist}((x_{j},P_{j}),(x^{\prime},P^{\prime}))<\eta_{(x_{j},P_{j})}/2.$

Thus, by definition, $(x^{\prime},P^{\prime})\in K_{j}$ and the conclusion
follows setting $\gamma=\min_{1\leq j\leq J}\tilde{\eta}_{(x_{j},P_{j})}/4$.

∎

######  Lemma 9.9.

There exists $0<\delta_{0}<1$ and a regular modulus of continuity ${\omega}$
for which the following holds:

Given $x,x^{\prime}\in E_{1}$ with $|x-x^{\prime}|\leq\delta_{0}$ and given
$(x,P)\in K$, there exists $P^{\prime}\in\Gamma(x^{\prime},2A^{\sharp})$ such
that

(9.40)
$|\partial^{\alpha}(P^{\prime}-P)(x)|\leq{\omega}(|x-x^{\prime}|)\cdot|x-x^{\prime}|^{m-|\alpha|}\text{
for }|\alpha|\leq m.$

###### Proof.

We mimic the proof of Lemma 6.4 in [6].

For $\nu=0,1,2,\cdots$, we set $\epsilon_{\nu}:=2^{-\nu}$. By Lemma 9.7, $K$
is compact and we may apply Lemma 9.6 to successively pick
$\delta_{0},\delta_{1},\delta_{2},\cdots$, such that the following hold.

  1. ($\delta$-1)

$\delta_{0}=1$.

  2. ($\delta$-2)

$0<\delta_{\nu+1}<\frac{1}{2}\delta_{\nu}$.

  3. ($\delta$-3)

If $\nu\geq 1$, then given $x,x^{\prime}\in E_{1}$ with
$\left|x-x^{\prime}\right|\leq\delta_{\nu}$, and given $(x,P)\in K$, there
exists $P^{\prime}\in\Gamma(x^{\prime},2A^{\sharp})$ with

$\left|\partial^{\alpha}(P^{\prime}-P)(x)\right|\leq\frac{1}{2}\epsilon_{\nu}\left|x^{\prime}-x\right|^{m-\left|\alpha\right|}\text{
for }\left|\alpha\right|\leq m.$

We define a regular modulus of continuity ${\omega}(t)$ on $[0,1]$ by setting

${\omega}(0)=0,\,{\omega}(\delta_{\nu})=\epsilon_{\nu},\,\text{and
}{\omega}(t)\text{ linear on each }[\delta_{\nu+1},\delta_{\nu}]\text{ for
}\nu\geq 0.$

Now suppose $x,x^{\prime}\in E_{1}$.

Suppose $x=x^{\prime}$, we may simply take $P=P^{\prime}$.

Suppose $x\neq x^{\prime}$. Furthermore, assume that
$0<\left|x-x^{\prime}\right|<\delta_{1}$. Let $(x,P)\in K$. Pick $\nu\geq 1$
such that $\delta_{\nu+1}<\left|x-x^{\prime}\right|\leq\delta_{\nu}$. By
($\delta$-3), there exists $P^{\prime}\in\Gamma(x^{\prime},2A^{\sharp})$ such
that

(9.41)
$\left|\partial^{\alpha}(P^{\prime}-P)(x)\right|\leq\frac{1}{2}\epsilon_{\nu}\left|x^{\prime}-x\right|^{{m-\left|\alpha\right|}}\text{
for }\left|\alpha\right|\leq m.$

On the other hand, since $\left|x^{\prime}-x\right|>\delta_{\nu+1}$, we have

(9.42)
${\omega}(\left|x^{\prime}-x\right|)\geq{\omega}(\delta_{\nu+1})=\epsilon_{\nu+1}=\frac{1}{2}\epsilon_{\nu}.$

Combining (9.41) and (9.42), we have

$\left|\partial^{\alpha}(P^{\prime}-P)(x)\right|\leq{\omega}(\left|x^{\prime}-x\right|)\left|x-x^{\prime}\right|^{{m-\left|\alpha\right|}}\text{
for }\left|\alpha\right|\leq m.$

∎

######  Lemma 9.10 (Existence of Modulus of Continuity).

Suppose

(9.43) ${k^{\sharp}}\geq D+2,1+(D+1){\tilde{k}}\leq k^{\sharp}$

and let $\omega$ be as in Lemma 9.9. Then, given any ${\overline{k}}\geq 1$,
there exists a controlled constant $\hat{C}_{{\overline{k}}}$ and
$\delta^{\prime}>0$ such that the following holds:

Let $x_{0}\in S\subset E_{1}$ with $\text{diam}(S)\leq\delta^{\prime}$ and
$|S|\leq\overline{k}$. Then, there exists a map $x\to P^{x}$ from $S$ into
$\mathcal{P}$ such that

(9.44) $P^{x}\in\Gamma(x,{\tilde{k}},CA^{\sharp}),(x,P^{x})\in K\text{ for
each }x\in S;\text{ and}$ (9.45)
$|\partial^{\alpha}(P^{x}-P^{y})(y)|\leq\hat{C}_{{\overline{k}}}\omega(|x-y|)|x-y|^{m-|\alpha|}\text{
for }x,y\in S,|x-y|\leq\delta^{\prime},|\alpha|\leq m.$

###### Proof.

In what follows, we first construct the mapping $x\mapsto P^{x}$ via Lemma 6.2
and repeated applications of Lemma 9.9 as in [6]. Then, we will show that each
step of the construction is valid in the sense that it produces $P^{x}$ such
that $(x,P^{x})\in K$, justifying the uses of Lemma 9.9 and satisfying 9.44.

The proof is by induction on ${\overline{k}}$, the base case
${\overline{k}}=1$ following from simply taking $P^{x_{0}}=P_{0}$ as
guaranteed by Lemma 9.8.

Now suppose Lemma 9.10 holds for all positive integers less than
${\overline{k}}$.

First require that $\delta_{1}<\delta_{0}$, where $\delta_{0}$ is as in Lemma
9.9.

By Lemma 6.2, we may partition $S$ into sets $S_{0},...,S_{M}$ such that

(9.46) $\\#(S_{l})\leq{\overline{k}}-1\text{ for each }l,(0\leq l\leq
M),\text{ and}$ (9.47)
$\text{dist}(S_{l},S_{l^{\prime}})>c_{{\overline{k}}}\cdot\text{diam}(S)\text{
for }l\neq l^{\prime}.$

Without loss of generality, take $x_{0}=x_{0}\in S_{0}$ and suppose each
$S_{l}$ is nonempty, fixing $x_{l}\in S_{l}$ ($l\geq 1$).