# Definable Lipschitz selections for affine-set valued maps

Adam Parusiński and Armin Rainer  Adam Parusiński: Université Côte d’Azur,
CNRS, LJAD, UMR 7351, 06108 Nice, France<EMAIL_ADDRESS>Armin Rainer: Fakultät für Mathematik, Universität Wien, Oskar-Morgenstern-
Platz 1, A-1090 Wien, Austria<EMAIL_ADDRESS>
###### Abstract.

Whitney’s extension problem, i.e., how one can tell whether a function
$f:X\to\mathbb{R}$, $X\subseteq\mathbb{R}^{n}$, is the restriction of a
$C^{m}$-function on $\mathbb{R}^{n}$, was solved in full generality by Charles
Fefferman in 2006. In this paper, we settle the $C^{1,\omega}$-case of a
related conjecture: given that $f$ is semialgebraic and $\omega$ is a
semialgebraic modulus of continuity, if $f$ is the restriction of a
$C^{1,\omega}$-function then it is the restriction of a semialgebraic
$C^{1,\omega}$-function. We work in the more general setting of sets that are
definable in an o-minimial expansion of the real field. An ingenious argument
of Brudnyi and Shvartsman relates the existence of $C^{1,\omega}$-extensions
to the existence of Lipschitz selections of certain affine-set valued maps. We
show that if a definable affine-set valued map has Lipschitz selections then
it also has definable Lipschitz selections. In particular, we obtain a
Lipschitz solution (more generally, $\omega$-Hölder solution, for any
definable modulus of continuity $\omega$) of the definable
Brenner–Epstein–Hochster–Kollár problem. In most of our results we have
control over the respective (semi)norms.

###### Key words and phrases:

O-minimal structures, Whitney’s extension problem, linear systems, Lipschitz
selections

###### 2020 Mathematics Subject Classification:

03C64, 14P10, 26B35, 26E25, 32B20, 46E15

Supported by FWF-Project P 32905-N and Oberwolfach Research Fellows (OWRF) ID
2244p

##  1\. Introduction

In this paper, we settle the $C^{1,\omega}$-case of an open problem raised by
Bierstone and Milman (cf. [21]): _Given a compact semialgebraic subset
$X\subseteq\mathbb{R}^{n}$ and a semialgebraic function $f:X\to\mathbb{R}$
which is the restriction of a $C^{m}$-function on $\mathbb{R}^{n}$, does there
exist a semialgebraic $C^{m}$-extension of $f$ to $\mathbb{R}^{n}$?_

For general $m$, the answer to this question is known to be affirmative in
dimension $n\leq 2$, due to Fefferman and Luli [10]. In arbitrary dimension,
Aschenbrenner and Thamrongthanyalak [2] proved the $C^{1}$-version of the
statement. A solution with loss of regularity is due to Bierstone, Campesato,
and Milman [5].

The philosophy of our approach is related to Aschenbrenner and
Thamrongthanyalak’s who prove and use a definable version of Michael’s
selection theorem. “Definable” means that the sets and maps belong to a fixed
o-minimal expansion of the real field (of which semialgebraic sets are a basic
example). Michael’s theorem concerns the existence of continuous selections of
set-valued maps. Already the existence of definable Lipschitz selections is in
general not known, since the classical results are based on transcendental
methods (e.g. the Steiner point).

Nevertheless, for certain affine-set valued maps, that are relevant for the
$C^{1,\omega}$-Whitney extension problem, Lipschitz selections can be
constructed in a way that preserves definability in the given o-minimal
structure. This construction is due to Brudnyi and Shvartsman [7]; see also
the references therein for precursors.

Let us now describe our results in more detail.

###  1.1. Main results

Let an o-minimal expansion of the real field be fixed. Throughout the paper, a
set $X\subseteq\mathbb{R}^{n}$ is called _definable_ if it is definable in
this fixed o-minimal structure. A map $\varphi:X\to\mathbb{R}^{m}$ is
definable if its graph $\\{(x,\varphi(x)):x\in X\\}$ is a definable subset of
$\mathbb{R}^{n}\times\mathbb{R}^{m}$. Cf. Section 2.1.

By a _modulus of continuity_ we always mean a positive, continuous,
increasing, and concave function $\omega:(0,\infty)\to(0,\infty)$ such that
$\omega(t)\to 0$ as $t\to 0$. A modulus of continuity $\omega$ is called
_definable_ if the function $\omega$ is definable.

By definition, $C^{1,\omega}(\mathbb{R}^{n})$ is the space of all
$C^{1}$-functions $f:\mathbb{R}^{n}\to\mathbb{R}$ that are globally bounded on
$\mathbb{R}^{n}$ and whose partial derivatives of first order are globally
bounded and globally $\omega$-Hölder on $\mathbb{R}^{n}$. Equipped with its
natural norm, this space is a Banach space. We write
$C^{1,\omega}_{\operatorname{def}}(\mathbb{R}^{n})$ for the subspace of
definable functions in $C^{1,\omega}(\mathbb{R}^{n})$. Given a definable
subset $X\subseteq\mathbb{R}^{n}$, we denote by
$C^{1,\omega}(\mathbb{R}^{n})|_{X}$ and
$C^{1,\omega}_{\operatorname{def}}(\mathbb{R}^{n})|_{X}$ the respective trace
spaces on $X$. Let $\mathbb{R}^{X}_{\operatorname{def}}$ be the set of all
definable functions $f:X\to\mathbb{R}$. For precise definitions we refer to
Section 2.2.

We will prove the following theorem.

######  Theorem A.

Let $\omega$ be a definable modulus of continuity. Let
$X\subseteq\mathbb{R}^{n}$ be a closed definable set and $f:X\to\mathbb{R}$ a
definable function. Then the following conditions are equivalent.

  1. (1)

$f$ is the restriction of a $C^{1,\omega}$-function on $\mathbb{R}^{n}$.

  2. (2)

$f$ is the restriction of a definable $C^{1,\omega}$-function on
$\mathbb{R}^{n}$.

That means

(1.1) $\mathbb{R}^{X}_{\operatorname{def}}\cap
C^{1,\omega}(\mathbb{R}^{n})|_{X}=C^{1,\omega}_{\operatorname{def}}(\mathbb{R}^{n})|_{X}.$

Moreover, a subset of the set (1.1) is bounded in
$C^{1,\omega}(\mathbb{R}^{n})|_{X}$ if and only if it is bounded in
$C^{1,\omega}_{\operatorname{def}}(\mathbb{R}^{n})|_{X}$.

In the Lipschitz case $\omega(t)=t$, for compact definable $X$ we even have

(1.2)
$\|f\|_{C^{1,1}_{\operatorname{def}}(\mathbb{R}^{n})|_{X}}\approx\|f\|_{C^{1,1}(\mathbb{R}^{n})|_{X}}.$

In particular, (1.1) holds for all classical Hölder classes with rational
Hölder exponent, since $\omega(t)=t^{\alpha}$ with rational $\alpha\in(0,1]$
is semialgebraic. We conjecture that

(1.3)
$\|f\|_{C^{1,\omega}_{\operatorname{def}}(\mathbb{R}^{n})|_{X}}\approx\|f\|_{C^{1,\omega}(\mathbb{R}^{n})|_{X}}$

for _any_ definable modulus of continuity $\omega$.

The meaning of “$\approx$” in (1.2) and (1.3) is that either quotient of the
two sides lies in the interval $[C^{-1},C]$ for some constant $C\geq 1$
depending only on $n$.

The proof of A is based on an ingenious argument of Brudnyi and Shvartsman [7]
that relates the existence of $C^{1,\omega}$-extensions to the existence of
Lipschitz selections for certain maps which take affine subspaces of
$\mathbb{R}^{n}$ as values.

The next result, B, states that, if these maps happen to be definable and
admit a Lipschitz selection, then they have a definable Lipschitz selection.

We denote by $\mathcal{A}_{k}(\mathbb{R}^{n})$ the set of all affine subspaces
of $\mathbb{R}^{n}$ of dimension at most $k$. A pseudometric space
$(\mathcal{M},\rho)$ is said to be definable if $\mathcal{M}$ is a definable
subset of some $\mathbb{R}^{N}$ and the pseudometric
$\rho:\mathcal{M}\times\mathcal{M}\to[0,\infty)$ is a definable function. A
map $F:\mathcal{M}\to\mathcal{A}_{k}(\mathbb{R}^{n})$ is called definable if
its graph $\bigcup_{m\in\mathcal{M}}(\\{m\\}\times F(m))$ is a definable
subset of $\mathbb{R}^{N}\times\mathbb{R}^{n}$. A _selection_ of $F$ is a map
$f:\mathcal{M}\to\mathbb{R}^{n}$ such that $f(m)\in F(m)$ for all
$m\in\mathcal{M}$. Cf. Section 2.3 and Section 2.4.

######  Theorem B.

Let $(\mathcal{M},\rho)$ be a definable pseudometric space and
$F:\mathcal{M}\to\mathcal{A}_{k}(\mathbb{R}^{n})$ a definable map. Then the
following conditions are equivalent.

  1. (1)

$F$ has a Lipschitz selection.

  2. (2)

$F$ has a definable Lipschitz selection.

Moreover, if there is a Lipschitz selection $f$ of $F$ with Lipschitz seminorm
$|f|_{\operatorname{Lip}(\mathcal{M},\mathbb{R}^{n})}$, then there is a
definable Lipschitz selection $g$ of $F$ with

(1.4) $|g|_{\operatorname{Lip}(\mathcal{M},\mathbb{R}^{n})}\leq
C(k,n)\,|f|_{\operatorname{Lip}(\mathcal{M},\mathbb{R}^{n})},$

where $C(k,n)>0$ is a constant that only depends on $k$ and $n$.

To deduce A from B we actually need a slightly stronger version of the latter
which is stated in Theorem 3.3.

Another application of B is an $\omega$-Hölder (in particular, Lipschitz)
solution of the definable Brenner–Epstein–Hochster–Kollár problem.

######  Theorem C.

Let $\omega$ be a definable modulus of continuity. Let
$A_{ij},b_{i}:X\to\mathbb{R}$, for $i=1,\ldots,N$ and $j=1,\ldots,M$, be
definable functions defined on a definable subset $X\subseteq\mathbb{R}^{n}$.
Consider the linear system of equations

(1.5) $\sum_{j=1}^{M}A_{ij}f_{j}=b_{i},\quad i=1,\ldots,N,$

in the unkowns $f_{j}$, $j=1,\ldots,M$. Then the following conditions are
equivalent.

  1. (1)

The system (1.5) admits an $\omega$-Hölder solution.

  2. (2)

The system (1.5) admits a definable $\omega$-Hölder solution.

Moreover, if there is an $\omega$-Hölder solution $f=(f_{1},\ldots,f_{M})$ of
(1.5) with $\omega$-Hölder seminorm $|f|_{C^{0,\omega}(X,\mathbb{R}^{M})}$,
then there is a definable $\omega$-Hölder solution $g=(g_{1},\ldots,g_{M})$ of
(1.5) such that

(1.6) $|g|_{C^{0,\omega}(X,\mathbb{R}^{M})}\leq
C(M)\,|f|_{C^{0,\omega}(X,\mathbb{R}^{M})}.$

A $C^{0}$-version of this theorem is due to Aschenbrenner and
Thamrongthanyalak [2]; for the semialgebraic setting see also Fefferman and
Kollár [9].

Let us emphasize that, due to [7], the respective first conditions in A and B
are characterized by _finiteness principles_ :

  * •

An arbitrary function $f:X\to\mathbb{R}$, $X\subseteq\mathbb{R}^{n}$, belongs
to $C^{1,\omega}(\mathbb{R}^{n})|_{X}$ if and only if for each subset
$Y\subseteq X$ of cardinality $\\#Y\leq 3\cdot 2^{n-1}$ there is $F_{Y}\in
C^{1,\omega}(\mathbb{R}^{n})$ such that $F_{Y}=f$ on $Y$ and
$\sup_{Y}\|F_{Y}\|_{C^{1,\omega}(\mathbb{R}^{n})}<\infty$.

  * •

An arbitrary map $F:\mathcal{M}\to\mathcal{A}_{k}(\mathbb{R}^{n})$, where
$\mathcal{M}$ is any pseudometric space, admits a Lipschitz selection if and
only if for each subset $\mathcal{N}\subseteq\mathcal{M}$ of cardinality
$\\#\mathcal{N}\leq 2^{k+1}$ the restriction $F|_{\mathcal{N}}$ has a
Lipschitz selection $f_{\mathcal{N}}$ such that
$\sup_{\mathcal{N}}|f_{\mathcal{N}}|_{\operatorname{Lip}(\mathcal{N},\mathbb{R}^{n})}<\infty$.

As shown in [7], the cardinalities of the finite test sets cannot be reduced.
Obviously, we immediately get a corresponding finiteness principle for C.

The proofs of the main results, A and B, are essentially a careful
verification that the constructions of [7] are done in a definable way. We
first prove B in Section 3 and then deduce A from it in Section 4. C is a
simple consequence of B; see Section 3.4.

For the transition from a definable Whitney jet of class $C^{1,\omega}$ on $X$
to a definable $C^{1,\omega}$-function on $\mathbb{R}^{n}$, we use our recent
paper [14] on the uniform extension of definable Whitney jets of class
$C^{m,\omega}$ (which is a variation of the definable $C^{m}$ Whitney
extension theorem [11, 12, 18]). At this stage, we cannot control the norms in
this extension and thus are not able to prove (1.3), but we prove that the
extension can be done in a bounded way; see also Section 4.4. Alternatively,
in the $C^{1,1}$-case one can use the explicit formula due to Azaga, Le
Gruyer, and Mudarra [3] which is visibly definable and admits a control of the
norms which leads to (1.2). The explicit formula also yields an immediate
proof of the definable Kirszbraun theorem; see Section 4.3.

### Notation

We equip $\mathbb{R}^{n}$ with the maximum-norm $\|x\|:=\max_{1\leq i\leq
n}|x_{i}|$. If a different norm is used, e.g., the Euclidean norm
$\|x\|_{2}:=(\sum_{1\leq i\leq n}x_{i}^{2})^{1/2}$, then it will be explicitly
stated. Note that $\|x\|\leq\|x\|_{2}\leq\sqrt{n}\,\|x\|$. The closed
$\|\cdot\|$-balls $Q(x,r):=\\{y\in\mathbb{R}^{n}:\|x-y\|\leq r\\}$ are cubes
with sides parallel to the coordinate axes. We also write $Q(r):=Q(0,r)$ so
that $Q(x,r)=x+Q(r)$. If $\lambda>0$ then $\lambda Q(x,r)$ denotes the cube
$x+Q(\lambda r)$. Then standard scalar product in $\mathbb{R}^{n}$ is denoted
by $\langle x,y\rangle:=\sum_{i=1}^{n}x_{i}y_{i}$.

##  2\. Preliminaries

In this section, we recall definitions and fix notation.

###  2.1. O-minimal expansions of the real field

An _o-minimal expansion of the ordered field of real numbers_ is a family
$\mathscr{S}=(\mathscr{S}_{n})_{n\geq 1}$, where $\mathscr{S}_{n}$ is a
collection of subsets of $\mathbb{R}^{n}$ such that

  * •

$\mathscr{S}_{n}$ is a boolean algebra with respect to the usual set-theoretic
operations,

  * •

$\mathscr{S}_{n}$ contains all semialgebraic subsets of $\mathbb{R}^{n}$,

  * •

$\mathscr{S}$ is stable by cartesian products and linear projections,

  * •

each $S\in\mathscr{S}_{1}$ has only finitely many connected components.

A set $S$ that belongs to $\mathscr{S}$ is said to be _definable_ (in
$\mathscr{S}$). A map $f:S\to\mathbb{R}^{m}$, $S\subseteq\mathbb{R}^{n}$, is
called _definable_ if its graph $\\{(x,f(x)):x\in S\\}$ is a definable subset
of $\mathbb{R}^{n}\times\mathbb{R}^{m}$.

The basic example of an o-mimimal expansion of the real field is the family of
semialgebraic sets. Another important example is the family of globally
subanalytic sets. Many more interesting o-minimal structures have been
identified in recent decades. We refer to [19] and [20] for the fundamentals
of the theory.

From now on, the attribute “definable” will refer to a fixed o-minimal
expansion of the real field.

###  2.2. Spaces of differentiable functions

Let $\omega$ be a modulus of continuity, i.e., a positive, continuous,
increasing, and concave function $\omega:(0,\infty)\to(0,\infty)$ such that
$\omega(t)\to 0$ as $t\to 0$. Let $C^{0,\omega}(\mathbb{R}^{n})$ be the set of
all continuous bounded functions $f:\mathbb{R}^{n}\to\mathbb{R}$ such that

$|f|_{C^{0,\omega}(\mathbb{R}^{n})}:=\inf\\{C>0:|f(x)-f(y)|\leq
C\,\omega(\|x-y\|)\text{ for all }x,y\in\mathbb{R}^{n}\\}<\infty.$

For $m$ a nonnegative integer, $C^{m,\omega}(\mathbb{R}^{n})$ consists of all
$C^{m}$-functions such that $\partial^{\alpha}f$ is globally bounded on
$\mathbb{R}^{n}$, for all $|\alpha|\leq m$, and $\partial^{\alpha}f\in
C^{0,\omega}(\mathbb{R}^{n})$, for all $|\alpha|=m$. Then
$C^{m,\omega}(\mathbb{R}^{n})$ is a Banach space with the norm

(2.1)
$\|f\|_{C^{m,\omega}(\mathbb{R}^{n})}:=\sup_{x\in\mathbb{R}^{n}}\sup_{|\alpha|\leq
m}|\partial^{\alpha}f(x)|+\sup_{|\alpha|=m}|\partial^{\alpha}f|_{C^{0,\omega}(\mathbb{R}^{n})}.$

Let $C^{m,\omega}_{\operatorname{def}}(\mathbb{R}^{n})$ be the subspace
consisting of the functions in $C^{m,\omega}(\mathbb{R}^{n})$ that are
definable.

Let $X\subseteq\mathbb{R}^{n}$ be a subset. The trace space
$C^{m,\omega}(\mathbb{R}^{n})|_{X}$ is the set of all $f:X\to\mathbb{R}$ such
that there exists $F\in C^{m,\omega}(\mathbb{R}^{n})$ with $F|_{X}=f$. It
carries the norm

$\|f\|_{C^{m,\omega}(\mathbb{R}^{n})|_{X}}:=\inf\\{\|F\|_{C^{m,\omega}(\mathbb{R}^{n})}:F\in
C^{m,\omega}(\mathbb{R}^{n}),\,F|_{X}=f\\}.$

Similarly, we consider the space
$C^{m,\omega}_{\operatorname{def}}(\mathbb{R}^{n})|_{X}$ of restrictions to
$X$ of functions in $C^{m,\omega}_{\operatorname{def}}(\mathbb{R}^{n})$ with
the norm

$\|f\|_{C^{m,\omega}_{\operatorname{def}}(\mathbb{R}^{n})|_{X}}:=\inf\\{\|F\|_{C^{m,\omega}(\mathbb{R}^{n})}:F\in
C^{m,\omega}_{\operatorname{def}}(\mathbb{R}^{n}),\,F|_{X}=f\\}.$

If $X$ is definable, then each element of
$C^{m,\omega}_{\operatorname{def}}(\mathbb{R}^{n})|_{X}$ is a definable
function on $X$.

###  2.3. Pseudometric spaces

By a _pseudometric space_ $(\mathcal{M},\rho)$ we mean a non-empty set
$\mathcal{M}$ together with a non-negative real valued function
$\rho:\mathcal{M}\times\mathcal{M}\to[0,\infty)$ such that $\rho(x,x)=0$,
$\rho(x,y)=\rho(y,x)$, and $\rho(x,y)\leq\rho(x,z)+\rho(z,y)$ for all
$x,y,z\in\mathcal{M}$. The function $\rho$ is called a _pseudometric_ ; if
additionally $\rho(x,y)=0$ implies $x=y$, it is called a _metric_. We say that
$(\mathcal{M},\rho)$ is an _extended pseudometric space_ if the pseudometric
$\rho$ may also take the value $+\infty$.

A pseudometric space $(\mathcal{M},\rho)$ is called _definable_ if
$\mathcal{M}$ is a definable subset of $\mathbb{R}^{N}$, for some $N$, and
$\rho:\mathcal{M}\times\mathcal{M}\to[0,\infty)$ is a definable function.

A map $f:\mathcal{M}\to\mathbb{R}^{n}$ is _Lipschitz_ if

$|f|_{\operatorname{Lip}(\mathcal{M},\mathbb{R}^{n})}:=\inf\\{C>0:\|f(x)-f(y)\|\leq
C\,\rho(x,y)\text{ for all }x,y\in\mathcal{M}\\}<\infty.$

Let $\omega$ be a modulus of continuity. A map $f:X\to\mathbb{R}^{n}$ with
$X\subseteq\mathbb{R}^{N}$ is _$\omega$ -Hölder_ if

$|f|_{C^{0,\omega}(X,\mathbb{R}^{n})}:=|f|_{\operatorname{Lip}(X_{\omega},\mathbb{R}^{n})}<\infty,$

where $X_{\omega}$ is the space $X$ endowed with the metric
$\rho(x,y):=\omega(\|x-y\|)$.

###  2.4. Set-valued mappings

A _set-valued mapping_ is a map $F:X\to 2^{Y}$. The _graph_ of $F$ is the
subset $\bigcup_{x\in X}(\\{x\\}\times F(x))$ of $X\times Y$. A _selection_ of
$F$ is a map $f:X\to Y$ such that $f(x)\in F(x)$ for all $x\in X$.

If $X\subseteq\mathbb{R}^{N}$ is definable and $Y=\mathbb{R}^{n}$, then we say
that a set-valued map $F:X\to 2^{\mathbb{R}^{n}}$ is _definable_ if its graph
is a definable subset of $\mathbb{R}^{N}\times\mathbb{R}^{n}$.

##  3\. Definable Lipschitz selections

The purpose of this section is to formulate and prove Theorem 3.3 from which B
and C will follow easily.

###  3.1. Definable Lipschitz selections for cube-valued maps

Let $\mathcal{Q}(\mathbb{R}^{n})$ be the family of all closed cubes
$Q(x,r):=\\{y\in\mathbb{R}^{n}:\|x-y\|\leq r\\}$, where $r\in[0,\infty]$; thus
$Q(x,0)=\\{x\\}$ and $Q(r,\infty)=\mathbb{R}^{n}$ count as cubes and belong to
$\mathcal{Q}(\mathbb{R}^{n})$. Cubes centered at the origin will be denoted by
$Q(r):=Q(0,r)$.

######  Lemma 3.1.

Let $(\mathcal{M},\rho)$ be a definable pseudometric space. Let
$F:\mathcal{M}\to\mathcal{Q}(\mathbb{R}^{n})$ be a definable map. Assume that
for every $2$-point subset $\mathcal{N}\subseteq\mathcal{M}$ there exists a
Lipschitz selection $f_{\mathcal{N}}:\mathcal{N}\to\mathbb{R}^{n}$ of
$F|_{\mathcal{N}}$ with
$|f_{\mathcal{N}}|_{\operatorname{Lip}(\mathcal{N},\mathbb{R}^{n})}\leq 1$.
Then there exists a definable Lipschitz selection
$f:\mathcal{M}\to\mathbb{R}^{n}$ of $F$ with
$|f|_{\operatorname{Lip}(\mathcal{M},\mathbb{R}^{n})}\leq 1$.

###### Proof.

Cf. [6, Proposition 1.24 and Corollary 1.25]. By projection to the coordinate
axes of $\mathbb{R}^{n}$ it is enough to consider the case $n=1$. Then $F(m)$
is a closed interval $[a(m),b(m)]$ (either bounded, possibly a point, or
$\mathbb{R}$).

Let us first show that we may assume that all $F(m)$ are bounded. Consider the
definable set $\mathcal{M}_{0}:=\\{m\in\mathcal{M}:F(m)\text{ is
bounded}\\}=b^{-1}(\mathbb{R})$ and the definable map
$F_{0}:=F|_{\mathcal{M}_{0}}$. We may assume that $\mathcal{M}_{0}$ is non-
empty, since otherwise $F(m)=\mathbb{R}$ for all $m\in\mathcal{M}$ and $f:=0$
is the required Lipschitz selection. If we prove that
$F_{0}:\mathcal{M}_{0}\to\mathcal{Q}(\mathbb{R})$ has a definable Lipschitz
selection $f_{0}:\mathcal{M}_{0}\to\mathbb{R}$ with
$|f_{0}|_{\operatorname{Lip}(\mathcal{M}_{0},\mathbb{R})}\leq 1$, then $f_{0}$
admits a definable Lipschitz extension $f:\mathcal{M}\to\mathbb{R}$ with
$|f|_{\operatorname{Lip}(\mathcal{M},\mathbb{R})}\leq 1$ defined by

(3.1)
$f(m):=\inf_{m^{\prime}\in\mathcal{M}_{0}}(f_{0}(m^{\prime})+\rho(m,m^{\prime})),\quad
m\in\mathcal{M},$

which clearly is the required selection of $F$ (since $F(m)=\mathbb{R}$ if
$m\not\in\mathcal{M}_{0}$). Let us check that $f$ in (3.1) has the desired
properties. It is finite, because

$f_{0}(m_{0})-\rho(m,m_{0})\leq f(m)\leq f_{0}(m_{0})+\rho(m,m_{0})$

for any $m_{0}\in\mathcal{M}_{0}$, where the first inequality holds, since
$|f_{0}|_{\operatorname{Lip}(\mathcal{M}_{0},\mathbb{R})}\leq 1$ entails

$f_{0}(x)+\rho(x,m)\geq f_{0}(m_{0})-\rho(x,m_{0})+\rho(x,m)\geq
f_{0}(m_{0})-\rho(m,m_{0}),$

for all $x\in\mathcal{M}_{0}$. We have $f|_{\mathcal{M}_{0}}=f_{0}$, since if
$m\in\mathcal{M}_{0}$ then

$f(m)\leq f_{0}(m)\leq f_{0}(m^{\prime})+\rho(m,m^{\prime})$

for all $m^{\prime}\in\mathcal{M}_{0}$, whence $f(m)=f_{0}(m)$. To see
$|f|_{\operatorname{Lip}(\mathcal{M},\mathbb{R})}\leq 1$ let
$m_{1},m_{2}\in\mathcal{M}$. Then

$f(m_{1})\leq\inf_{m^{\prime}\in\mathcal{M}_{0}}(f_{0}(m^{\prime})+\rho(m_{2},m^{\prime})+\rho(m_{1},m_{2}))=f(m_{2})+\rho(m_{1},m_{2}),$

and we are done.

Hence, it suffices to prove the lemma under the assumption that $F(m)$ is
bounded for all $m\in\mathcal{M}$. We will show that

$f(m):=\inf_{m^{\prime}\in\mathcal{M}}(b(m^{\prime})+\rho(m,m^{\prime}))=\inf_{m^{\prime}\in\mathcal{M}}(\sup
F(m^{\prime})+\rho(m,m^{\prime})),\quad m\in\mathcal{M},$

is a definable Lipschitz selection of $F$ with
$|f|_{\operatorname{Lip(\mathcal{M},\mathbb{R})}}\leq 1$; in particular, $f$
is finite. It is immediate from the definition that $f$ is definable and
satisfies $f(m)\leq b(m)$ for every $m\in\mathcal{M}$. To see $a(m)\leq f(m)$
let $m^{\prime}\in\mathcal{M}$ be arbitrary. By assumption, there exists
$g:=f_{\\{m,m^{\prime}\\}}:\\{m,m^{\prime}\\}\to\mathbb{R}$ with $a(m)\leq
g(m)\leq b(m)$, $a(m^{\prime})\leq g(m^{\prime})\leq b(m^{\prime})$, and
$|g(m)-g(m^{\prime})|\leq\rho(m,m^{\prime})$, so that $a(m)\leq
b(m^{\prime})+\rho(m,m^{\prime})$.

Finally, for any $m_{1},m_{2}\in\mathcal{M}$,

$f(m_{1})\leq\inf_{m^{\prime}\in\mathcal{M}}(b(m^{\prime})+\rho(m_{2},m^{\prime})+\rho(m_{1},m_{2}))=f(m_{2})+\rho(m_{1},m_{2}),$

and we conclude that $|f|_{\operatorname{Lip}(\mathcal{M},\mathbb{R})}\leq 1$.
∎

###  3.2. Definable Lipschitz selections for affine-set valued maps

Let $V\subseteq\mathbb{R}^{N}$ be a definable set. Let $\Gamma=(V,E)$ be a
graph with set of vertices $V$ and set of edges
$E\subseteq\\{(v,v^{\prime})\in V\times V:v\neq v^{\prime}\\}$. We assume that
$E$ is symmetric in the sense that $(v,v^{\prime})\in E$ implies
$(v^{\prime},v)\in E$. In other words, the graph is undirected and, strictly
speaking, the set of edges actually is $E/\\!\sim$, where
$(v,v^{\prime})\sim(v^{\prime},v)$, but this inaccuracy in notation will cause
no troubles. Furthermore, we assume that $E$ is definable. Let us write
$v\leftrightarrow v^{\prime}$ if $v,v^{\prime}\in V$ are joined by an edge,
i.e., $(v,v^{\prime})\in E$.

We say that a subset $W\subseteq V$ is _admissible_ if $(W,E_{W})$ regarded as
a subgraph of $\Gamma=(V,E)$, where $(v,v^{\prime})\in E_{W}$ if and only if
$(v,v^{\prime})\in E$ and $v,v^{\prime}\in W$, has no isolated vertices. This
means that each vertex is joined by an edge to another vertex unless the graph
consists of but one vertex.

Let us endow the graph $\Gamma=(V,E)$ with a _weight_ , i.e., a symmetric
function $w:E\to[0,\infty]$. This induces an extended pseudometric space
$(V,\sigma)$, where $\sigma:V\times V\to[0,\infty]$ is defined by

(3.2) $\sigma(v,v^{\prime}):=\inf\sum_{i=0}^{k}w(e_{i}),\quad v\neq
v^{\prime},$

where the infimum is taken over all finite paths $\\{e_{i}\\}_{i=0}^{k}$,
$k\in\mathbb{N}$, of edges in $\Gamma$ joining $v$ and $v^{\prime}$. Moreover,
$\sigma(v,v):=0$ and $\sigma(v,v^{\prime}):=+\infty$ if there is no path of
edges joining $v\neq v^{\prime}$.

We say that $\Gamma=(V,E,w)$ is a _definable weighted graph_ if the sets $V$
and $E$ are definable as specified above and, additionally, there exist a
definable pseudometric $\rho:V\times V\to[0,\infty)$ on $V$ and an absolute
constant $A\geq 1$ such that

(3.3) $\frac{1}{A}\rho\leq\sigma\leq A\rho.$

So $(V,\sigma)$ is a pseudometric space, while $(V,\rho)$ is even a definable
pseudometric space. In particular, in a definable weighted graph any two
vertices are joined by a path of edges and $V$ is an admissible subset of $V$.

######  Remark 3.2.

Let $(\mathcal{M},\rho)$ be a definable pseudometric space. The full graph
with set of vertices $\mathcal{M}$, set of edges
$\\{(m,m^{\prime})\in\mathcal{M}\times\mathcal{M}:m\neq m^{\prime}\\}$, and
weight $\rho$ is a definable weighted graph (where $\sigma=\rho$).
Furthermore, any subset of $\mathcal{M}$ is admissible.

Recall that $\mathcal{A}_{k}(\mathbb{R}^{n})$ is the family of affine
subspaces of $\mathbb{R}^{n}$ of dimension at most $k$ and that a map
$F:V\to\mathcal{A}_{k}(\mathbb{R}^{n})$ is called definable if its graph is a
definable subset of $\mathbb{R}^{N}\times\mathbb{R}^{n}$. For a map
$f:W\to\mathbb{R}^{n}$, $W\subseteq V$, we set

$|f|_{\operatorname{Lip}(W,\mathbb{R}^{n})}:=\inf\\{C>0:\|f(v)-f(v^{\prime})\|\leq
C\,\rho(v,v^{\prime})\text{ for all }v,v^{\prime}\in W\\};$

i.e., the Lipschitz seminorm is computed in terms of $\rho$ (not $\sigma$).

######  Theorem 3.3.

Let $\Gamma=(V,E,w)$ be a definable weighted graph. Let
$F:V\to\mathcal{A}_{k}(\mathbb{R}^{n})$ be a definable map. Assume that for
each admissible subset $W\subseteq V$ of cardinality $\\#W\leq 2^{k+1}$ there
exists a Lipschitz selection $f_{W}:W\to\mathbb{R}^{n}$ of $F|_{W}$ with
$|f_{W}|_{\operatorname{Lip}(W,\mathbb{R}^{n})}\leq 1$. Then there exists a
definable Lipschitz selection $f:V\to\mathbb{R}^{n}$ of $F$ with
$|f|_{\operatorname{Lip}(V,\mathbb{R}^{n})}\leq C$, where $C\geq 1$ is a
constant depending only on $k$, $n$, and $A$ from (3.3).

Note that Theorem 3.3 implies B in view of Remark 3.2. It is a definable
version of [7, Theorem 2.3]. We closely follow its proof making sure that all
steps are definable.

Let us give a rough outline of the general strategy of the proof.

$\textstyle{\boxed{(V,\rho)\stackrel{{\scriptstyle
F}}{{\longrightarrow}}\mathcal{A}_{k+1}(\mathbb{R}^{n})}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(1)}$$\textstyle{\boxed{(\hat{V},\hat{\rho})\stackrel{{\scriptstyle\hat{F}}}{{\longrightarrow}}\mathcal{Q}(\mathbb{R}^{n})}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(3)}$$\textstyle{\boxed{(\overline{V},\overline{\rho})\stackrel{{\scriptstyle\overline{F}}}{{\longrightarrow}}\mathcal{A}_{k}(\mathbb{R}^{n})}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(2)}$

It is by induction on $k$. The induction step ($k\to k+1$) is divided into the
following three stages (which correspond to the bent arrows in the diagram).

  1. (1)

From $F:V\to\mathcal{A}_{k+1}(\mathbb{R}^{n})$ one constructs a “doubling”
$(\overline{V},\overline{\rho})$ of the space $(V,\rho)$ and a map
$\overline{F}:\overline{V}\to\mathcal{A}_{k}(\mathbb{R}^{n})$ in such a way
that the induction hypothesis yields a Lipschitz selection $\overline{f}$ of
$\overline{F}$.

  2. (2)

The Lipschitz selection $\overline{f}$ of $\overline{F}$ is used to define a
new space $(\hat{V}=\overline{V},\hat{\rho})$ and a cube-valued map
$\hat{F}:\hat{V}\to\mathcal{Q}(\mathbb{R}^{n})$; the center of the cube
$\hat{F}(\overline{v})$ is $\overline{f}(\overline{v})$. Thanks to Lemma 3.1,
there is a Lipschitz selection $\hat{f}$ of $\hat{F}$ which moreover can be
interpreted as a Lipschitz map defined on $(V,\rho)$.

  3. (3)

The desired Lipschitz selection $f$ of $F$ is finally found by defining $f(v)$
to be the orthogonal projection of $\hat{f}(v)$ to the affine subspace $F(v)$
of $\mathbb{R}^{n}$.

###  3.3. Proof of Theorem 3.3

We prove Theorem 3.3 by induction on $k$.

The base case $k=0$ is trivial:
$\mathcal{A}_{0}(\mathbb{R}^{n})\cong\mathbb{R}^{n}$ so that
$f=F:V\to\mathbb{R}^{n}$ is the desired definable Lipschitz selection. Indeed,
let $v,v^{\prime}\in V$ be any two distinct vertices. Then there is a path of
edges $v=:v_{0}\leftrightarrow v_{1}\leftrightarrow\cdots\leftrightarrow
v_{\ell}:=v^{\prime}$ joining $v$ and $v^{\prime}$ so that, by assumption (as
$\\{v_{i},v_{i+1}\\}$ is admissible) and (3.3),

(3.4) $\|f(v)-f(v^{\prime})\|\leq\sum_{i=0}^{\ell-1}\rho(v_{i},v_{i+1})\leq
A\sum_{i=0}^{\ell-1}w(v_{i},v_{i+1}).$

Taking the infimum over all paths joining $v$ and $v^{\prime}$ and using again
(3.3), we conclude that $|f|_{\operatorname{Lip}(V,\mathbb{R}^{n})}\leq
A^{2}$.

Let us assume that the result holds for some $k<n$ and prove it for $k+1$. Let
$F:V\to\mathcal{A}_{k+1}(\mathbb{R}^{n})$ be a definable map and assume that
for each admissible subset $W\subseteq V$ of cardinality $\\#W\leq 2^{k+2}$
there exists a Lipschitz selection $f_{W}:W\to\mathbb{R}^{n}$ of $F|_{W}$ with
$|f_{W}|_{\operatorname{Lip}(W,\mathbb{R}^{n})}\leq 1$.

Let $v_{1},v_{2}\in V$ be such that $v_{1}\leftrightarrow v_{2}$. Then
$\\{v_{1},v_{2}\\}$ is an admissible subset of $V$ and the assumption implies
that there exist points $x_{1}\in F(v_{1})$, $x_{2}\in F(v_{2})$ such that

(3.5) $\|x_{1}-x_{2}\|\leq\rho(v_{1},v_{2}).$

It follows that the set

$\\{(x_{1},x_{2})\in F(v_{1})\times
F(v_{2}):\|x_{1}-x_{2}\|\leq\rho(v_{1},v_{2})\\}$

is non-empty and definable, since $F$ and $\rho$ are definable. Therefore, by
definable choice, we may assume that $x_{i}=x_{i}(v_{1},v_{2})$, $i=1,2$,
depend in a definable way on $v_{1}$ and $v_{2}$. We may conclude that

(3.6)
$P(v_{1},v_{2}):=F(v_{1})\cap\big{(}F(v_{2})+Q(2\rho(v_{1},v_{2}))+x_{1}(v_{1},v_{2})-x_{2}(v_{1},v_{2})\big{)}$

is a non-empty convex definable set that is symmetric with respect to the
point $x_{1}$.

Let us understand better the geometry of $P(v_{1},v_{2})$. For ease of
notation let $U_{1}:=F(v_{1})$ and $U_{2}:=F(v_{2})+x_{1}-x_{2}$. Then $U_{1}$
and $U_{2}$ are affine subspaces of $\mathbb{R}^{n}$ through $x_{1}$ of
dimension at most $k+1$. Set $d_{i}:=\dim U_{i}$, $i=1,2$. Moreover, the
“slab” $S_{2}:=U_{2}+Q(2\rho(v_{1},v_{2}))$ is the thickening of $U_{2}$ by
the cube $Q(2\rho(v_{1},v_{2}))$.

If $d_{1}=0$, then

$\displaystyle P(v_{1},v_{2})=U_{1}\cap
S_{2}=\\{x_{1}\\}=U_{1}\cap\big{(}\\{x_{1}\\}+Q(0)\big{)}.$

Now assume that $d_{1}>0$. If $U_{1}\cap U_{2}=\\{x_{1}\\}$, then
$P(v_{1},v_{2})$ is a polytope in $U_{1}$ centered at $x_{1}$ with pairwise
parallel faces. In fact, the pairs of parallel faces result as the
intersections with $U_{1}$ of opposite parallel faces of the slab $S_{2}$. Let
$p$ be the number of pairs of parallel faces of $P(v_{1},v_{2})$. Thus $p\leq
n$. (Note that $p=n$ may occur if $d_{2}=0$ so that $S_{2}$ is a cube.) Let
$L_{1},\ldots,L_{p}$ be the affine subspaces of $U_{1}$ of dimension $d_{1}-1$
through $x_{1}$ that are equidistant to the respective affine hulls of
opposite parallel faces of $P(v_{1},v_{2})$, the distance being computed with
respect to $\|\cdot\|$ and having the respective values $r_{1},\ldots,r_{p}$.
Then

(3.7)
$P(v_{1},v_{2})=\bigcap_{i=1}^{p}\big{(}U_{1}\cap(L_{i}+Q(r_{i}))\big{)}.$

Now suppose that $U_{1}\cap U_{2}\neq\\{x_{1}\\}$. If $U_{1}\cap U_{2}\neq
U_{1}$, then $P(v_{1},v_{2})$ is a polytope in $U_{1}$ centered at $x_{1}$
with pairwise parallel faces and infinite extension in the directions of
$U_{1}\cap U_{2}$. The number of pairs of parallel faces $p$ again satisfies
$p\leq n$. Let $L_{1},\ldots,L_{p}$ be the affine subspaces of $U_{1}$ of
dimension $d_{1}-1$ through $x_{1}$ that are equidistant to the respective
affine hulls of _finite_ opposite parallel faces of $P(v_{1},v_{2})$. As
before the distance is computed with respect to $\|\cdot\|$ and let
$r_{1},\ldots,r_{p}$ be the respective values. Then we again have the
representation (3.7).

It remains to consider the case $U_{1}\cap U_{2}=U_{1}$. Then
$P(v_{1},v_{2})=U_{1}$. If $d_{1}\leq k$, we set $L_{1}:=U_{1}$ and $r_{1}:=0$
so that (3.7) remains valid with $p=1$. If $d_{1}=k+1$, we define
$L_{1}:=\\{x_{1}\\}$ and $r_{1}:=\infty$. Again (3.7) holds true with $p=1$.
Note that this is the only case, where we allow an infinite radius. This case
occurs if and only if $F(v_{1})$ and $F(v_{2})$ are parallel and both have
dimension $k+1$.

Thus we proved that in any case

(3.8) $P(v_{1},v_{2})=\bigcap_{i\in
I(v_{1},v_{2})}\big{(}F(v_{1})\cap(L_{i}+Q(r_{i}))\big{)},$

where the index set $I(v_{1},v_{2})$ has cardinality at least $1$ and at most
$n$ and where each $L_{i}$ is an affine subspace of $F(v_{1})$ through $x_{1}$
of dimension at most $k$. The radii $r_{i}$ are finite unless
$F(v_{1})\|F(v_{2})$ and $\dim F(v_{1})=\dim F(v_{2})=k+1$.

Let us consider the definable sets

$\overline{V}_{\|}:=\\{(v_{1},v_{2}):v_{1}\leftrightarrow
v_{2},\,F(v_{1})\|F(v_{2}),\,\dim F(v_{1})=\dim F(v_{2})=k+1\\}.$

and

$\overline{V}_{0}:=\\{\overline{v}=(v_{1},v_{2},i):v_{1}\leftrightarrow
v_{2},\,(v_{1},v_{2})\not\in\overline{V}_{\|},\,i\in I(v_{1},v_{2})\\}.$

We get a new definable pseudometric space
$(\overline{V}_{0},\overline{\rho})$, where, for
$\overline{v}=(v_{1},v_{2},i)\neq\overline{v}^{\prime}=(v_{1}^{\prime},v_{2}^{\prime},i^{\prime})$,

$\overline{\rho}(\overline{v},\overline{v}^{\prime}):=\rho(v_{1},v_{1}^{\prime})+r_{i}+r_{i^{\prime}}.$

We extend this data to an extended pseudometric space
$(\overline{V},\overline{\rho})$ by defining

$\overline{V}:=\\{\overline{v}=(v_{1},v_{2},i):v_{1}\leftrightarrow
v_{2},\,i\in I(v_{1},v_{2})\\}$

and putting $\overline{\rho}(\overline{v},\overline{v}^{\prime}):=+\infty$ if
at least one of $\overline{v}\neq\overline{v}^{\prime}$ does not belong to
$\overline{V}_{0}$.

Define $\overline{F}:\overline{V}\to\mathcal{A}_{k}(\mathbb{R}^{n})$ by
setting $\overline{F}(v_{1},v_{2},i):=L_{i}$. By the above discussion,
$\overline{F}$ is definable.

######  Claim 1.

There exists a definable Lipschitz selection
$\overline{f}:\overline{V}\to\mathbb{R}^{n}$ of $\overline{F}$ with
$|\overline{f}|_{\operatorname{Lip}(\overline{V},\mathbb{R}^{n})}\leq C(k,n)$.

We will first consider the restriction
$\overline{F}_{0}:=\overline{F}|_{\overline{V}_{0}}$. By Remark 3.2, we may
view $(\overline{V}_{0},\overline{\rho})$ as a definable weighted graph, where
each subset of vertices is admissible (and $A=1$). Thus, by the induction
hypothesis, it suffices to show that for every subset
$\overline{W}\subseteq\overline{V}_{0}$ of cardinality $\\#\overline{W}\leq
2^{k+1}$ there is a Lipschitz selection
$\overline{f}_{\overline{W}}:\overline{W}\to\mathbb{R}^{n}$ of
$\overline{F}|_{\overline{W}}$ with
$|\overline{f}_{\overline{W}}|_{\operatorname{Lip}(\overline{W},\mathbb{R}^{n})}\leq
1$.

For $\overline{v}=(v_{1},v_{2},i)$ we write

$\operatorname{pr}_{1}(\overline{v})=v_{1}(\overline{v})=v_{1},\quad\operatorname{pr}_{2}(\overline{v})=v_{2}(\overline{v})=v_{2},\quad\operatorname{pr}_{3}(\overline{v})=i(\overline{v})=i.$

Set
$W:=\operatorname{pr}_{1}\overline{W}\cup\operatorname{pr}_{2}\overline{W}$.
Then $W$ is an admissible subset of $V$ of cardinality $\\#W\leq
2\overline{W}\leq 2^{k+2}$, since for each $v\in W$ there is $v^{\prime}\in W$
such that $v\leftrightarrow v^{\prime}$. Thus, by the assumption on $F$, there
exists a Lipschitz selection $f_{W}:W\to\mathbb{R}^{n}$ of $F|_{W}$ with
$|f_{W}|_{\operatorname{Lip}(W,\mathbb{R}^{n})}\leq 1$.

Let $v_{1},v_{2}\in W$. Then $f_{W}(v_{1})\in F(v_{1})$ and, by (3.5),

$\displaystyle\|f_{W}(v_{1})-f_{W}(v_{2})-x_{1}(v_{1},v_{2})+x_{2}(v_{1},v_{2})\|$
$\displaystyle\leq\|f_{W}(v_{1})-f_{W}(v_{2})\|+\|x_{1}(v_{1},v_{2})-x_{2}(v_{1},v_{2})\|$
$\displaystyle\leq 2\,\rho(v_{1},v_{2}).$

Consequently, $f_{W}(v_{1})\in P(v_{1},v_{2})$; cf. (3.6).

Now define $\overline{f}_{\overline{W}}:\overline{W}\to\mathbb{R}^{n}$ by
letting $\overline{f}_{\overline{W}}(\overline{v})$ be a point in
$\overline{F}(\overline{v})=L_{i(\overline{v})}$ which minimizes the
$\|\cdot\|$-distance to $f_{W}(v_{1}(\overline{v}))$. Since
$f_{W}(v_{1}(\overline{v}))\in P(v_{1}(\overline{v}),v_{2}(\overline{v}))$,
(3.8) implies that $f_{W}(v_{1}(\overline{v}))\in
L_{i(\overline{v})}+Q(r_{i(\overline{v})})$. Then (by the minimality property
of $\overline{f}_{\overline{W}}(\overline{v})$) we have

$\|\overline{f}_{\overline{W}}(\overline{v})-f_{W}(v_{1}(\overline{v}))\|\leq
r_{i(\overline{v})},$

and consequently

$\displaystyle\|\overline{f}_{\overline{W}}(\overline{v})-\overline{f}_{\overline{W}}(\overline{v}^{\prime})\|$
$\displaystyle\leq\|f_{W}(v_{1}(\overline{v}))-f_{W}(v_{1}(\overline{v}^{\prime}))\|+r_{i(\overline{v})}+r_{i(\overline{v}^{\prime})}$
$\displaystyle\leq\rho(v_{1}(\overline{v})),v_{1}(\overline{v}^{\prime}))+r_{i(\overline{v})}+r_{i(\overline{v}^{\prime})}$
$\displaystyle=\overline{\rho}(\overline{v},\overline{v}^{\prime}).$

Thus,
$|\overline{f}_{\overline{W}}|_{\operatorname{Lip}(\overline{W},\mathbb{R}^{n})}\leq
1$.

By the induction hypothesis, we may infer that there is a definable Lipschitz
selection $\overline{f}_{0}:\overline{V}_{0}\to\mathbb{R}^{n}$ of
$\overline{F}_{0}$ with
$|\overline{f}_{0}|_{\operatorname{Lip}(\overline{V}_{0},\mathbb{R}^{n})}\leq
C(k,n)$.

Let us extend $\overline{f}_{0}$ to a map
$\overline{f}:\overline{V}\to\mathbb{R}^{n}$ by setting

$\overline{f}(\overline{v}):=\begin{cases}\overline{f}_{0}(\overline{v})&\text{
if }\overline{v}\in\overline{V}_{0},\\\ x_{1}(\overline{v})&\text{ if
}\overline{v}\not\in\overline{V}_{0}.\end{cases}$

Then $\overline{f}$ is obviously a definable selection of $\overline{F}$.
Evidently,
$|\overline{f}|_{\operatorname{Lip}(\overline{V},\mathbb{R}^{n})}\leq C(k,n)$,
since $\overline{\rho}(\overline{v},\overline{v}^{\prime})=\infty$ whenever at
least one of $\overline{v}\neq\overline{v}^{\prime}$ does not belong to
$\overline{V}_{0}$.

Thus 1 is proved.

######  Claim 2.

There is a definable Lipschitz map $\hat{f}:\overline{V}\to\mathbb{R}^{n}$
such that

  1. (i)

$\hat{f}(\overline{v})$ depends only on $v_{1}(\overline{v})$; thus we may
regard $\hat{f}$ as a map defined on $V$,

  2. (ii)

$|\hat{f}|_{\operatorname{Lip}(V,\mathbb{R}^{n})}\leq C$,

  3. (iii)

$\|\hat{f}(\overline{v})-\overline{f}(\overline{v})\|\leq
C\,r_{i(\overline{v})}$ for all $\overline{v}\in\overline{V}$,

where $C=C(k,n)$ is the constant from 1.

In order to define $\hat{f}$, we consider the definable pseudometric space
$(\hat{V},\hat{\rho})$, where $\hat{V}:=\overline{V}$ and

$\hat{\rho}(\overline{v},\overline{v}^{\prime}):=C\,\rho(v_{1}(\overline{v}),v_{1}(\overline{v}^{\prime})).$

Let $\hat{F}:\hat{V}\to\mathcal{Q}(\mathbb{R}^{n})$ be defined by
$\hat{F}(\overline{v}):=Q(\overline{f}(\overline{v}),Cr_{i(\overline{v})})$.
Then $\hat{F}$ is definable. We will show that $\hat{F}$ has a definable
Lipschitz selection $\hat{f}:\hat{V}\to\mathbb{R}^{n}$ such that
$|\hat{f}|_{\operatorname{Lip}(\hat{V},\mathbb{R}^{n})}\leq 1$. Then 2 follows
easily: we have

$\|\hat{f}(\overline{v})-\hat{f}(\overline{v}^{\prime})\|\leq\hat{\rho}(\overline{v},\overline{v}^{\prime})=C\,\rho(v_{1}(\overline{v}),v_{1}(\overline{v}^{\prime}))$

so that $\hat{f}(\overline{v})=\hat{f}(\overline{v}^{\prime})$ if
$v_{1}(\overline{v})=v_{1}(\overline{v}^{\prime})$ which implies (i) and (ii).
Since $\hat{f}$ is a selection of $\hat{F}$, (iii) is clear.

Let $\overline{v},\overline{v}^{\prime}\in\hat{V}=\overline{V}$ be any two
distinct points. By 1,

$\|\overline{f}(\overline{v})-\overline{f}(\overline{v}^{\prime})\|\leq
C\,\overline{\rho}(\overline{v},\overline{v}^{\prime})=C\,r_{i(\overline{v})}+C\,r_{i(\overline{v}^{\prime})}+C\,\rho(v_{1}(\overline{v}),v_{1}(\overline{v}^{\prime})).$

This means that the cubes $\hat{F}(\overline{v})$ and
$\hat{F}(\overline{v}^{\prime})$ have $\|\cdot\|$-distance at most
$C\,\rho(v_{1}(\overline{v}),v_{1}(\overline{v}^{\prime}))$. So there exist
points $y\in\hat{F}(\overline{v})$ and
$y^{\prime}\in\hat{F}(\overline{v}^{\prime})$ such that

$\|y-y^{\prime}\|\leq
C\,\rho(v_{1}(\overline{v}),v_{1}(\overline{v}^{\prime}))=\hat{\rho}(\overline{v},\overline{v}^{\prime}).$

In other words, for every $2$-point subset
$\\{\overline{v},\overline{v}^{\prime}\\}$ of $\hat{V}$ the restriction
$\hat{F}|_{\\{\overline{v},\overline{v}^{\prime}\\}}$ has a Lipschitz
selection $\hat{f}_{\\{\overline{v},\overline{v}^{\prime}\\}}$ with
$|\hat{f}_{\\{\overline{v},\overline{v}^{\prime}\\}}|_{\operatorname{Lip}(\\{\overline{v},\overline{v}^{\prime}\\},\mathbb{R}^{n})}\leq
1$. By Lemma 3.1, there is a definable Lipschitz selection
$\hat{f}:\hat{V}\to\mathbb{R}^{n}$ of $\hat{F}$ with
$|\hat{f}|_{\operatorname{Lip}(\hat{V},\mathbb{R}^{n})}\leq 1$. This ends the
proof of 2.

######  Claim 3.

The desired definable Lipschitz selection $f:V\to\mathbb{R}^{n}$ of $F$ with
$|f|_{\operatorname{Lip}(V,\mathbb{R}^{n})}\leq C_{1}(k,n,A)$ is given by

$f(v):=\operatorname{pr}_{F(v)}\hat{f}(v),$

i.e., $f(v)$ is the orthogonal projection of $\hat{f}(v)$ to the affine
subspace $F(v)$ of $\mathbb{R}^{n}$.

It is clear that $f$ is a definable selection of $F$. To check the Lipschitz
property we show that

(3.9) $\|f(v)-f(v^{\prime})\|\leq C_{1}\,\rho(v,v^{\prime}),\quad\text{
whenever }v\leftrightarrow v^{\prime},$

where $C_{1}=C_{1}(k,n)$. This suffices in view of (3.2) and (3.3) (and a
computation similar to (3.4) after which $C_{1}$ will also depend on $A$).

In order to prove (3.9), we need some technical facts. First of all, we may
assume that $\dim F(v)\geq\dim F(v^{\prime})$; if not we just interchange the
roles of $v$ and $v^{\prime}$.

######  Claim 4.

The cube
$K_{v}:=\bigcap\\{Q(f(v),r_{i(\overline{v})}):\overline{v}\in\overline{V},\,\operatorname{pr}_{1}(\overline{v})=v\\}$
satisfies

$\hat{f}(v)\in\lambda K_{v},$

where $\lambda:=(1+\sqrt{n})C$ and $C$ is the constant from 1, and

$K_{v}\cap F(v)\subseteq
F(v^{\prime})+Q(2\lambda\rho(v,v^{\prime}))+x_{1}(v,v^{\prime})-x_{2}(v,v^{\prime})$

whenever $v\leftrightarrow v^{\prime}$. (Recall that $\lambda K_{v}$ denotes
the cube with the same center as $K_{v}$ and $\lambda$ times its radius.)

Since $\overline{v}:=(v,v^{\prime},i)\in\overline{V}$, for $i\in
I(v,v^{\prime})$, we have, by 2(iii),

$\|\hat{f}(v)-\overline{f}(\overline{v})\|\leq C\,r_{i(\overline{v})}.$

As
$\overline{f}(\overline{v})\in\overline{F}(\overline{v})=L_{i(\overline{v})}\subseteq
F(v)$, the definition of $f(v)$ implies

$\displaystyle\|f(v)-\overline{f}(\overline{v})\|$
$\displaystyle\leq\|f(v)-\overline{f}(\overline{v})\|_{2}$
$\displaystyle\leq\|\hat{f}(v)-\overline{f}(\overline{v})\|_{2}\leq\sqrt{n}\,\|\hat{f}(v)-\overline{f}(\overline{v})\|\leq\sqrt{n}\,C\,r_{i(\overline{v})}.$

These inequalities easily yield $\hat{f}(v)\in\lambda K_{v}$.

If $x\in K_{v}$, then

$\|x-f(v)\|\leq r_{i(\overline{v})},\quad\text{ for all
}\overline{v}\in\overline{V}\text{ such that
}\operatorname{pr}_{1}(\overline{v})=v,$

whence

$\|x-\overline{f}(\overline{v})\|\leq\lambda r_{i(\overline{v})}.$

Thus, and by the fact that
$I(v,v^{\prime})=\\{i(\overline{v}):\overline{v}\in\overline{V},\,\operatorname{pr}_{1}(\overline{v})=v,\,\operatorname{pr}_{2}(\overline{v})=v^{\prime}\\}$,

$K_{v}\subseteq\bigcap_{i\in
I(v,v^{\prime})}Q(\overline{f}(\overline{v}),\lambda
r_{i})\subseteq\bigcap_{i\in I(v,v^{\prime})}\big{(}L_{i}+Q(\lambda
r_{i})\big{)}.$

Intersecting both sides with $F(v)$ and using (3.6) and (3.8) (modulo a
dilation with center $x_{1}(v,v^{\prime})$ by the factor $\lambda$), also the
second assertion in 4 follows.

Let us continue with the proof of 3. By 4, there exists $y(v,v^{\prime})\in
F(v^{\prime})+x_{1}(v,v^{\prime})-x_{2}(v,v^{\prime})$ with

(3.10) $\|f(v)-y(v,v^{\prime})\|\leq 2\lambda\,\rho(v,v^{\prime}).$

We consider the translate $F(v,v^{\prime})$ of $F(v^{\prime})$ through $f(v)$
defined by

$F(v,v^{\prime}):=F(v^{\prime})+x_{1}(v,v^{\prime})-x_{2}(v,v^{\prime})-y(v,v^{\prime})+f(v).$

Let us prove (3.9). For $v\leftrightarrow v^{\prime}$, we have

$\displaystyle\|f(v)-f(v^{\prime})\|$
$\displaystyle\leq\|\operatorname{pr}_{F(v)}\hat{f}(v)-\operatorname{pr}_{F(v,v^{\prime})}\hat{f}(v)\|$
$\displaystyle\quad+\|\operatorname{pr}_{F(v,v^{\prime})}\hat{f}(v)-\operatorname{pr}_{F(v,v^{\prime})}\hat{f}(v^{\prime})\|$
$\displaystyle\quad+\|\operatorname{pr}_{F(v,v^{\prime})}\hat{f}(v^{\prime})-\operatorname{pr}_{F(v^{\prime})}\hat{f}(v^{\prime})\|.$

The second term on the right-hand side is bounded by

$\|\hat{f}(v)-\hat{f}(v^{\prime})\|_{2}\leq\sqrt{n}\,\|\hat{f}(v)-\hat{f}(v^{\prime})\|\leq
C\sqrt{n}\,\rho(v,v^{\prime}),$

thanks to 2. The third term is bounded by

$\|x_{1}(v,v^{\prime})-x_{2}(v,v^{\prime})-y(v,v^{\prime})+f(v)\|_{2}\leq\sqrt{n}(1+2\lambda)\,\rho(v,v^{\prime}),$

in view of (3.5) and (3.10).

It remains to estimate the first term. By 4 and (3.10),

$K_{v}\cap F(v)\subseteq F(v,v^{\prime})+Q(4\lambda\,\rho(v,v^{\prime})),$

and, by a dilation with center $f(v)$ and factor $\sqrt{n}\,\lambda$, we find

$\big{(}\sqrt{n}\,\lambda K_{v}\big{)}\cap F(v)\subseteq
F(v,v^{\prime})+Q(4\sqrt{n}\,\lambda^{2}\rho(v,v^{\prime})).$

Let $B$ be the biggest Euclidean ball with center $f(v)$ that is contained in
$\sqrt{n}\,\lambda K_{v}$. Then

$\lambda K_{v}\subseteq B\subseteq\sqrt{n}\,\lambda K_{v}.$

Let $B(r)$ be the Euclidean ball with center $0$ and radius
$r:=4n\lambda^{2}\rho(v,v^{\prime})$. Then

$B\cap F(v)\subseteq F(v,v^{\prime})+B(r),$

and since the orthogonal projection of a point in $B\cap F(v)$ on
$F(v,v^{\prime})$ is contained in $B\cap F(v,v^{\prime})$, we have

$B\cap F(v)\subseteq B\cap F(v,v^{\prime})+B(r).$

Recall that we assumed that $\dim F(v)\geq\dim F(v^{\prime})=\dim
F(v,v^{\prime})$ (see the paragraph before 4). Since both these affine spaces
pass through the center $f(v)$ of $B$, we thus also have

$B\cap F(v,v^{\prime})\subseteq B\cap F(v)+B(r).$

That means that for the Hausdorff distance of $B\cap F(v)$ and $B\cap
F(v,v^{\prime})$ we have

$d_{\mathcal{H}}(B\cap F(v),B\cap F(v,v^{\prime}))\leq
r=4n\lambda^{2}\rho(v,v^{\prime}).$

Now it suffices to invoke [7, Lemma 2.9] (note that is uses
$\hat{f}(v)\in\lambda K_{v}$ shown in 4) which gives that

$\|\operatorname{pr}_{F(v)}\hat{f}(v)-\operatorname{pr}_{F(v,v^{\prime})}\hat{f}(v)\|\leq
d_{\mathcal{H}}(B\cap F(v),B\cap F(v,v^{\prime})).$

This completes the proof of 3 and thus the proof of Theorem 3.3.

###  3.4. Proof of C

Let us define

$F(x):=\Big{\\{}(f_{1},\ldots,f_{M})\in\mathbb{R}^{M}:\sum_{j=1}^{M}A_{ij}(x)f_{j}=b_{i}(x),\,i=1,\ldots,N\Big{\\}},\quad
x\in X.$

We assume that the system (1.5) has a solution that is $\omega$-Hölder. In
particular, $F(x)$ is a non-empty affine subspace of $\mathbb{R}^{M}$ for each
$x\in X$. Since the functions $A_{ij}$ and $b_{i}$ are assumed definable on
$X$, we see that $F:X\to\mathcal{A}_{M}(\mathbb{R}^{M})$ is a definable map.
Now B (applied to $\mathcal{M}=X$ with the metric $\rho(x,y)=\omega(\|x-y\|)$)
shows that the system (1.5) has a definable $\omega$-Hölder solution. Also the
statement (1.6) on the $\omega$-Hölder seminorms is immediate from B.

##  4\. Definable $C^{1,\omega}$ extension of functions

We will now work towards the proof of A. First, we will establish a
correspondence between definable Whitney jets of class $C^{1,\omega}$ and
definable Lipschitz selections of a special affine-set valued map. It will
allow us to bring to bear Theorem 3.3.

###  4.1. Definable Whitney jets vs. definable Lipschitz selections

Let $\omega$ be a modulus of continuity. We will henceforth assume that

(4.1) $\omega\leq 1.$

This is no restriction regarding A, since replacing $\omega$ by
$\overline{\omega}:=\min\\{1,\omega\\}$ in (2.1) gives equivalent norms:

$\|f\|_{C^{m,\omega}(\mathbb{R}^{n})}\leq\|f\|_{C^{m,\overline{\omega}}(\mathbb{R}^{n})}\leq
3\,\|f\|_{C^{m,\omega}(\mathbb{R}^{n})}.$

The first inequality is immediate from $\overline{\omega}\leq\omega$. For the
second inequality, it is enough to consider $m=0$ and show that
$|f|_{C^{0,\overline{\omega}}(\mathbb{R}^{n})}\leq
2\,\|f\|_{C^{0,\omega}(\mathbb{R}^{n})}$. To this end let
$t_{0}:=\inf\\{t>0:\omega(t)\geq 1\\}$. If $\|x-y\|<t_{0}$, then

$|f(x)-f(y)|\leq|f|_{C^{0,\omega}(\mathbb{R}^{n})}\,\omega(\|x-y\|)=|f|_{C^{0,\omega}(\mathbb{R}^{n})}\,\overline{\omega}(\|x-y\|),$

and, if $\|x-y\|\geq t_{0}$,

$\frac{|f(x)-f(y)|}{\overline{\omega}(\|x-y\|)}=|f(x)-f(y)|\leq
2\sup_{x\in\mathbb{R}^{n}}|f(x)|,$

which implies the assertion.

Let $X$ be a closed definable subset of $\mathbb{R}^{n}$. Set

$\mathcal{M}_{X}:=\\{(x,y)\in X\times X:x\neq y\\}$

and endow it with the metric $\rho_{\omega}$ defined by

$\rho_{\omega}((x,y),(x^{\prime},y^{\prime})):=\omega(\|x-y\|)+\omega(\|x^{\prime}-y^{\prime}\|)+\omega(\|x-x^{\prime}\|),$

if $(x,y)\neq(x^{\prime},y^{\prime})$, and $0$ otherwise. (At this stage, we
do not assume that $\omega$ is definable, but eventually we will.)

Let $f:X\to\mathbb{R}$ be a definable function. Consider the definable map
$L_{f}:\mathcal{M}_{X}\to\mathcal{A}_{n-1}(\mathbb{R}^{n})$ defined by

$L_{f}(x,y):=\\{z\in\mathbb{R}^{n}:\langle z,x-y\rangle=f(x)-f(y)\\}.$

Let us show that $f$ can be completed to a definable Whitney jet of class
$C^{1,\omega}$ on $X$ if and only if $L_{f}$ has a definable Lipschitz
selection.

Recall that $(f,g)$ is a _definable Whitney jet of class $C^{1,\omega}$ on
$X$_ if $f:X\to\mathbb{R}$ and $g:X\to\mathbb{R}^{n}$ are definable maps and

$\displaystyle\|(f,g)\|_{X,1,\omega}$ $\displaystyle:=\sup_{x\in
X}|f(x)|+\sup_{x\in X}\|g(x)\|+|(f,g)|_{X,1,\omega}<\infty,$ where
$\displaystyle|(f,g)|_{X,1,\omega}$
$\displaystyle:=\sup_{\begin{subarray}{c}x,y\in X\\\ x\neq
y\end{subarray}}\frac{|f(x)-f(y)-\langle
g(y),x-y\rangle|}{\|x-y\|\,\omega(\|x-y\|)}+\sup_{\begin{subarray}{c}x,y\in
X\\\ x\neq y\end{subarray}}\frac{\|g(x)-g(y)\|}{\omega(\|x-y\|)}.$

######  Proposition 4.1.

Let $X\subseteq\mathbb{R}^{n}$ be a closed definable set and let
$f:X\to\mathbb{R}$ be a bounded definable function. Then the following
conditions are equivalent:

  1. (1)

There exists a bounded definable function $g:X\to\mathbb{R}^{n}$ such that
$(f,g)$ is a definable Whitney jet of class $C^{1,\omega}$ on $X$.

  2. (2)

There exists a bounded definable Lipschitz selection
$\ell:\mathcal{M}_{X}\to\mathbb{R}^{n}$ of $L_{f}$.

If the equivalent conditions 1 and 2 hold, then

(4.2) $\inf_{g}\|(f,g)\|_{X,1,\omega}\approx\sup_{x\in
X}|f(x)|+\inf_{\ell}\Big{\\{}\sup_{(x,y)\in\mathcal{M}_{X}}\|\ell(x,y)\|+|\ell|_{\operatorname{Lip}(\mathcal{M}_{X},\mathbb{R}^{n})}\Big{\\}},$

where the infimum on the left-hand side is taken over all $g$ satisfying 1 and
the infimum on the right-hand side over all $\ell$ satisfying 2.

Note that “$\approx$” in (4.2) means that any of the two sides is bounded by
the other side up to a multiplicative factor which depends only on $n$.

###### Proof.

(1) $\Rightarrow$ (2) Let $g$ be as in (1) so that $(f,g)$ is a definable
Whitney jet of class $C^{1,\omega}$ on $X$. For $(x,y)\in\mathcal{M}_{X}$ let
$\ell(x,y)$ be the point in $L_{f}(x,y)$ closest to $g(x)$ (with respect to
the Euclidean metric), i.e.,

(4.3) $\|g(x)-\ell(x,y)\|_{2}=\operatorname{dist}_{2}(g(x),L_{f}(x,y)).$

Then $\ell:\mathcal{M}_{X}\to\mathbb{R}^{n}$ is a definable selection of
$L_{f}$. Let us check that it is bounded and Lipschitz. We have

$\displaystyle\|g(x)-\ell(x,y)\|_{2}=\frac{|f(x)-f(y)-\langle
g(x),x-y\rangle|}{\|x-y\|_{2}}$

which is bounded by

$\displaystyle\|(f,g)\|_{X,1,\omega}\frac{\|x-y\|\,\omega(\|x-y\|)}{\|x-y\|_{2}}\leq\|(f,g)\|_{X,1,\omega},$

in view of $\omega\leq 1$ (cf. (4.1)). Thus

$\|\ell(x,y)\|\leq\|g(x)\|+\|g(x)-\ell(x,y)\|\leq 2\,\|(f,g)\|_{X,1,\omega},$

i.e., $\ell$ is bounded with $\sup_{(x,y)\in\mathcal{M}_{X}}\|\ell(x,y)\|\leq
2\,\|(f,g)\|_{X,1,\omega}$.

For $(x,y),(x^{\prime},y^{\prime})\in\mathcal{M}_{X}$, we have

$\displaystyle\|\ell(x,y)-\ell(x^{\prime},y^{\prime})\|$
$\displaystyle\leq\|\ell(x,y)-g(x)\|+\|g(x)-g(x^{\prime})\|+\|g(x^{\prime})-\ell(x^{\prime},y^{\prime})\|$
$\displaystyle\leq\|(f,g)\|_{X,1,\omega}\big{(}\omega(\|x-y\|)+\omega(\|x-x^{\prime}\|)+\omega(\|x^{\prime}-y^{\prime}\|)\big{)}$
$\displaystyle=\|(f,g)\|_{X,1,\omega}\,\rho_{\omega}((x,y),(x^{\prime},y^{\prime})),$

consequently,
$|\ell|_{\operatorname{Lip}(\mathcal{M}_{X},\mathbb{R}^{n})}\leq\|(f,g)\|_{X,1,\omega}$.

(2) $\Rightarrow$ (1) Assume that $L_{f}$ has a bounded definable Lipschitz
selection $\ell:\mathcal{M}_{X}\to\mathbb{R}^{n}$. For ease of notation, set

$C_{\ell}:=\sup_{(x,y)\in\mathcal{M}_{X}}\|\ell(x,y)\|+|\ell|_{\operatorname{Lip}(\mathcal{M}_{X},\mathbb{R}^{n})}.$

If $x\in X$ is an isolated point of $X$, let $\hat{x}$ be a closest point in
$X\setminus\\{x\\}$ (with respect to $\|\cdot\|$). Otherwise, let
$\hat{x}:=x$. The point $\hat{x}$ can be assigned to $x$ in a definable way.
Define $g:X\to\mathbb{R}^{n}$ by

$g(x):=\begin{cases}\ell(x,\hat{x})&\text{ if }x\text{ is an isolated point of
}X,\\\ \lim_{X\ni y\to x}\ell(x,y)&\text{ otherwise. }\end{cases}$

Then $g$ is definable and $\|g(x)\|\leq C_{\ell}$ for all $x\in X$. The limit
exists, since

$\|\ell(x,y)-\ell(x,y^{\prime})\|\leq
C_{\ell}\,\rho_{\omega}((x,y),(x,y^{\prime}))=C_{\ell}\big{(}\omega(\|x-y\|)+\omega(\|x-y^{\prime}\|)\big{)}.$

Let us check that $(f,g)$ is a Whitney jet of class $C^{1,\omega}$ on $X$. For
each $x\in X$ let $(x_{i})$ be a sequence in $X\setminus\\{x\\}$ (possibly
stationary) such that $x_{i}\to\hat{x}$ and $\ell(x,x_{i})\to g(x)$. Then

$\displaystyle\|g(x)-g(x^{\prime})\|$
$\displaystyle=\lim_{i\to\infty}\|\ell(x,x_{i})-\ell(x^{\prime},x^{\prime}_{i})\|$
$\displaystyle\leq
C_{\ell}\lim_{i\to\infty}\rho_{\omega}((x,x_{i}),(x^{\prime},x^{\prime}_{i}))$
$\displaystyle\leq
C_{\ell}\big{(}\omega(\|x-\hat{x}\|)+\omega(\|x^{\prime}-\hat{x}^{\prime}\|)+\omega(\|x-x^{\prime}\|)\big{)}\leq
3C_{\ell}\,\omega(\|x-x^{\prime}\|).$

Since $\ell(x,y)\in L_{f}(x,y)$, we have

$\displaystyle|f(x)-f(y)-\langle g(x),x-y\rangle|$
$\displaystyle=|\langle\ell(x,y)-g(x),x-y\rangle|$
$\displaystyle=\lim_{i\to\infty}|\langle\ell(x,y)-\ell(x,x_{i}),x-y\rangle|$
$\displaystyle\leq n\,\|x-y\|\limsup_{i\to\infty}\|\ell(x,y)-\ell(x,x_{i})\|$
$\displaystyle\leq
n\,C_{\ell}\,\|x-y\|\limsup_{i\to\infty}\rho_{\omega}((x,y),(x,x_{i}))$
$\displaystyle\leq
n\,C_{\ell}\,\|x-y\|\big{(}\omega(\|x-y\|)+\omega(\|x-\hat{x}\|)\big{)}$
$\displaystyle\leq 2n\,C_{\ell}\,\|x-y\|\,\omega(\|x-y\|).$

Thus $|(f,g)|_{X,1,\omega}\leq 2n\,C_{\ell}$. ∎

By a little trick, the boundedness condition for $\ell$ in Proposition 4.1 may
be absorbed by the Lipschitz condition. To this end, we add a point $*$ to
$\mathcal{M}_{X}$. We define
$\widetilde{\mathcal{M}}_{X}:=\mathcal{M}_{X}\cup\\{*\\}$ and extend the
metric to $\widetilde{\mathcal{M}}_{X}$ by setting
$\widetilde{\rho}_{\omega}|_{\mathcal{M}_{X}\times\mathcal{M}_{X}}:=\rho_{\omega}$
and $\widetilde{\rho}_{\omega}((x,y),*)=\widetilde{\rho}_{\omega}(*,(x,y)):=2$
for all $(x,y)\in\mathcal{M}_{X}$ as well as
$\widetilde{\rho}_{\omega}(*,*):=0$. (Since $\omega\leq 1$, the triangle
inequality holds: for $m,m^{\prime}\in\mathcal{M}_{X}$ we have
$\rho_{\omega}(m,m^{\prime})\leq 3\leq
2+2=\rho_{\omega}(m,*)+\rho_{\omega}(*,m^{\prime})$.)

The map $L_{f}$ is extended to
$\widetilde{L}_{f}:\widetilde{\mathcal{M}}_{X}\to\mathcal{A}_{n-1}(\mathbb{R}^{n})$
by setting $\widetilde{L}_{f}|_{\mathcal{M}_{X}}=L_{f}$ and
$\widetilde{L}_{f}(*):=\\{0\\}$.

######  Proposition 4.2.

Under the assumptions of Proposition 4.1, items 1 and 2 in that proposition
are further equivalent to

  1. (3)

There exists a definable Lipschitz selection
$\widetilde{\ell}:\widetilde{\mathcal{M}}_{X}\to\mathbb{R}^{n}$ of
$\widetilde{L}_{f}$.

If the equivalent conditions 1, 2, and 3 hold, then we also have

(4.4) $\inf_{g}\|(f,g)\|_{X,1,\omega}\approx\sup_{x\in
X}|f(x)|+\inf_{\widetilde{\ell}}|\widetilde{\ell}|_{\operatorname{Lip}(\widetilde{\mathcal{M}}_{X},\mathbb{R}^{n})},$

where the infimum on the left-hand side is taken over all $g$ satisfying 1 and
the infimum on the right-hand side over all $\widetilde{\ell}$ satisfying 3.

###### Proof.

To see that (3) implies (2), note that

$\frac{\|\widetilde{\ell}(x,y)\|}{2}=\frac{\|\widetilde{\ell}(x,y)-\widetilde{\ell}(*)\|}{\widetilde{\rho}_{\omega}((x,y),*)}\leq|\widetilde{\ell}|_{\operatorname{Lip}(\widetilde{\mathcal{M}}_{X},\mathbb{R}^{n})}$

for all $(x,y)\in\mathcal{M}_{X}$, if $\widetilde{\ell}$ is a Lipschitz
selection of $\widetilde{L}_{f}$.

Conversely, if $\ell$ is a bounded definable Lipschitz selection of $L_{f}$,
then the unique extension $\widetilde{\ell}|_{\mathcal{M}_{X}}:=\ell$ and
$\widetilde{\ell}(*):=0$ is a definable Lipschitz selection of
$\widetilde{L}_{f}$. Indeed, for $(x,y)\in\mathcal{M}_{X}$,

$\|\widetilde{\ell}(x,y)-\widetilde{\ell}(*)\|=\|\ell(x,y)\|\leq
C=\frac{C}{2}\,\widetilde{\rho}_{\omega}((x,y),*),$

where $C=\sup_{(x,y)\in\mathcal{M}_{X}}\|\ell(x,y)\|$. ∎

Next we endow $\widetilde{\mathcal{M}}_{X}$ with a weighted graph structure.
Let $\widetilde{\mathcal{M}}_{X}$ be the set of vertices of this graph. Two
vertices $(x,y),(x^{\prime},y^{\prime})\in\mathcal{M}_{X}$ are joined by an
edge if $\\{x,y\\}\cap\\{x^{\prime},y^{\prime}\\}\neq\emptyset$, and $*$ is
joined by an edge to every $(x,y)\in\mathcal{M}_{X}$. If
$m,m^{\prime}\in\widetilde{\mathcal{M}}_{X}$ are joined by an edge, then we
assign the weight

$w(m,m^{\prime}):=\begin{cases}\omega(\|x-y\|)+\omega(\|x^{\prime}-y^{\prime}\|)&\text{
if }m=(x,y),\,m^{\prime}=(x^{\prime},y^{\prime})\in\mathcal{M}_{X},\\\
2&\text{ otherwise. }\end{cases}$

Let $\sigma$ be the associated (extended) pseudometric; cf. (3.2). By [7,
Proposition 3.3] (and thanks to our general assumption $\omega\leq 1$; cf.
(4.1)), we have

(4.5) $\frac{1}{2}\widetilde{\rho}_{\omega}\leq\sigma\leq
2\widetilde{\rho}_{\omega}.$

######  Corollary 4.3.

If $\omega$ is definable, then $(\widetilde{M}_{X},\widetilde{\rho}_{\omega})$
is a definable metric space and, equipped with the above weighted graph
structure, it is a definable weighted graph.

###### Proof.

This follows from the definitions and (4.5). ∎

We recall a consequence of [7, Proposition 3.2 and Corollary 3.4] (which is an
analogue of Proposition 4.2, where sets and maps are not necessarily
definable).

######  Proposition 4.4 ([7, Proposition 3.5]).

Let $m\geq 1$ be an integer. Let $X\subseteq\mathbb{R}^{n}$ and
$f:X\to\mathbb{R}$ be given (not necessarily definable). Assume that the
restriction $f|_{Y}$ to every subset $Y\subseteq X$ of cardinality $\\#Y\leq
m$ has an extension $F_{Y}\in C^{1,\omega}(\mathbb{R}^{n})$ with
$\|F_{Y}\|_{C^{1,\omega}(\mathbb{R}^{n})}\leq 1$. If $\mathcal{N}$ is an
admissible subset of $\widetilde{\mathcal{M}}_{X}$ of cardinality
$\\#\mathcal{N}\leq\frac{2}{3}m$, then $\widetilde{L}_{f}|_{\mathcal{N}}$ has
a Lipschitz selection
$\widetilde{\ell}_{\mathcal{N}}:\mathcal{N}\to\mathbb{R}^{n}$ with
$|\widetilde{\ell}_{\mathcal{N}}|_{\operatorname{Lip}(\mathcal{N},\mathbb{R}^{n})}\leq
C(n)$.

We are ready for the goal of this section:

######  Theorem 4.5.

Let $\omega$ be a definable modulus of continuity. Let
$X\subseteq\mathbb{R}^{n}$ be a closed definable set and $f:X\to\mathbb{R}$ a
definable function. Assume that the restriction $f|_{Y}$ to every subset
$Y\subseteq X$ of cardinality $\\#Y\leq 3\cdot 2^{n-1}$ has an extension
$F_{Y}\in C^{1,\omega}(\mathbb{R}^{n})$ with
$\|F_{Y}\|_{C^{1,\omega}(\mathbb{R}^{n})}\leq 1$. Then there exists a bounded
definable function $g:X\to\mathbb{R}^{n}$ such that $(f,g)$ is a definable
Whitney jet of class $C^{1,\omega}$ on $X$ with $\|(f,g)\|_{X,1,\omega}\leq
C(n)$.

###### Proof.

Applying the assumption to $1$-point sets $Y\subseteq X$, we may conclude that
$\sup_{x\in X}|f(x)|\leq 1$. The assumption and Proposition 4.4 give that for
every admissible subset $\mathcal{N}\subseteq\widetilde{\mathcal{M}}_{X}$ of
cardinality $\\#\mathcal{N}\leq\frac{2}{3}\cdot 3\cdot 2^{n-1}=2^{n}$ there
exists a Lipschitz selection
$\widetilde{\ell}_{\mathcal{N}}:\mathcal{N}\to\mathbb{R}^{n}$ of
$\widetilde{L}_{f}|_{\mathcal{N}}$ with
$|\widetilde{\ell}_{\mathcal{N}}|_{\operatorname{Lip}(\mathcal{N},\mathbb{R}^{n})}\leq
C(n)$. By Theorem 3.3 and Corollary 4.3, there exists a definable Lipschitz
selection $\widetilde{\ell}:\widetilde{\mathcal{M}}_{X}\to\mathbb{R}^{n}$ of
$\widetilde{L}_{f}$ with
$|\widetilde{\ell}|_{\operatorname{Lip}(\widetilde{\mathcal{M}}_{X},\mathbb{R}^{n})}\leq
C_{1}(n)$; note that $k=n-1$ and $A=2$ in this case. Now the assertion follows
from Proposition 4.2. ∎

###  4.2. Proof of A

We shall see that the identity (1.1) and the statement about the bounded
subsets follow from Theorem 4.5 and the uniform definable $C^{m,\omega}$
Whitney extension theorem:

######  Theorem 4.6 ([14]).

Let $0\leq m\leq p$ be integers. Let $\omega$ be a modulus of continuity. Let
$X\subseteq\mathbb{R}^{n}$ be a definable closed set. Any definable bounded
family of Whitney jets of class $C^{m,\omega}$ on $X$ extends to a definable
bounded family of $C^{m,\omega}$-functions on $\mathbb{R}^{n}$ which are of
class $C^{p}$ outside $X$.

Clearly, boundedness is understood with respect to the natural norms. The main
theorem of [14] is actually more general.

Let us prove (1.1) and the statement about the bounded subsets. It is obvious
that
$C^{1,\omega}_{\operatorname{def}}(\mathbb{R}^{n})|_{X}\subseteq\mathbb{R}^{X}_{\operatorname{def}}\cap
C^{1,\omega}(\mathbb{R}^{n})|_{X}$ and

$\|f\|_{C^{1,\omega}(\mathbb{R}^{n})|_{X}}\leq\|f\|_{C^{1,\omega}_{\operatorname{def}}(\mathbb{R}^{n})|_{X}},\quad
f\in C^{1,\omega}_{\operatorname{def}}(\mathbb{R}^{n})|_{X}.$

Conversely, suppose that $f\in\mathbb{R}^{X}_{\operatorname{def}}\cap
C^{1,\omega}(\mathbb{R}^{n})|_{X}$. Then Theorem 4.5 implies that there is a
bounded definable function $g:X\to\mathbb{R}^{n}$ such that $(f,g)$ is a
definable Whitney jet of class $C^{1,\omega}$ on $X$ with

$\|(f,g)\|_{X,1,\omega}\leq C(n)\,\|f\|_{C^{1,\omega}(\mathbb{R}^{n})|_{X}}.$

By Theorem 4.6, we conclude that $f\in
C^{1,\omega}_{\operatorname{def}}(\mathbb{R}^{n})|_{X}$ and that a subset of
$\mathbb{R}^{X}_{\operatorname{def}}\cap C^{1,\omega}(\mathbb{R}^{n})|_{X}$
which is bounded in $C^{1,\omega}(\mathbb{R}^{n})|_{X}$ is also bounded in
$C^{1,\omega}_{\operatorname{def}}(\mathbb{R}^{n})|_{X}$.

Let us now specialize to the case $\omega(t)=t$ (we write
$\|\cdot\|_{X,1,1}:=\|\cdot\|_{X,1,\omega}$ and
$|\cdot|_{X,1,1}:=|\cdot|_{X,1,\omega}$ in this case) and prove (1.2). To this
end, consider:

######  Proposition 4.7.

Let $X\subseteq\mathbb{R}^{n}$ be a definable compact set and let
$f:X\to\mathbb{R}$ be a bounded definable function. Assume that
$\mathcal{M}_{X}$ carries the metric $\rho_{\omega}$ with $\omega(t)=t$. Then
the following conditions are equivalent:

  1. (1)

$f$ is the restriction of a definable function $F\in C^{1,1}(\mathbb{R}^{n})$.

  2. (2)

There exists a definable function $g:X\to\mathbb{R}^{n}$ such that $(f,g)$ is
a Whitney jet of class $C^{1,1}$ on $X$.

Moreover, if $g$ is as in 2, then there exists an extension $F$ of $f$ such
that

(4.6) $\displaystyle\|F\|_{C^{1,1}(\mathbb{R}^{n})}\leq
C(n)\,\|(f,g)\|_{X,1,1}.$

###### Proof.

Let us recall a result of Azagra, Le Gruyer, and Mudarra [3]:

_Let $(f,g)$ be a Whitney jet of class $C^{1,1}$ on a set
$X\subseteq\mathbb{R}^{n}$ and let $M>0$ be such that $|(f,g)|_{X,1,1}\leq M$.
Then_

$\displaystyle F$
$\displaystyle:=\operatorname{conv}(h)-\frac{\sqrt{n}\,M}{2}\|\cdot\|_{2}^{2},\quad{where}$
$\displaystyle h(x)$ $\displaystyle:=\inf_{y\in X}\Big{(}f(y)+\langle
g(y),x-y\rangle+\frac{\sqrt{n}\,M}{2}\|x-y\|_{2}^{2}\Big{)}+\frac{\sqrt{n}\,M}{2}\|x\|_{2}^{2},\quad
x\in\mathbb{R}^{n},$

_and where $\operatorname{conv}(h)$ is the convex envelope of $h$, defines a
$C^{1}$-function $F:\mathbb{R}^{n}\to\mathbb{R}$ such that $F|_{X}=f$, $\nabla
F|_{X}=g$, and $|\nabla
F|_{\operatorname{Lip}(\mathbb{R}^{n},\mathbb{R}^{n})}\leq n\,M$. _

Note that the factor $\sqrt{n}$ appears, since we change from Euclidean to
maximum norm. The extension $F$ is optimal; cf. [3, Theorem 3.4]. The convex
envelope $\operatorname{conv}(h)$ (i.e., the supremum of all convex, proper,
l.s.c. functions $\varphi\leq h$) can be expressed as

$\operatorname{conv}(h)(x)=\inf\Big{\\{}\sum_{j=1}^{n+1}\lambda_{j}h(x_{j}):x=\sum_{j=1}^{n+1}\lambda_{j}x_{j},\,\sum_{j=1}^{n+1}\lambda_{j}=1,\,\lambda_{j}\geq
0\Big{\\}}.$

or as $\operatorname{conv}(h)=(h^{*})^{*}$, where
$h^{*}(x):=\sup_{y\in\mathbb{R}^{n}}\big{(}\langle y,x\rangle-h(y)\big{)}$ is
the convex conjugate. Now, if $X$ and $(f,g)$ are definable, we see that also
the extension $F$ is definable (because it is given by definable formulas).
After multiplication with a suitable definable $C^{2}$-cutoff function that
equals $1$ in a small neighborhood of $X$ and vanishes outside the
$1$-neighborhood of $X$ (here we use that $X$ is compact), we get that $F$
belongs to $C^{1,1}(\mathbb{R}^{n})$. This shows that (2) implies (1). The
opposite direction is clear.

Now it is not hard to check (4.6). ∎

######  Theorem 4.8.

Let $X\subseteq\mathbb{R}^{n}$ be a definable compact set and
$f:X\to\mathbb{R}$ a definable function. Assume that the restriction $f|_{Y}$
to every subset $Y\subseteq X$ of cardinality $\\#Y\leq 3\cdot 2^{n-1}$ has an
extension $F_{Y}\in C^{1,1}(\mathbb{R}^{n})$ with
$\|F_{Y}\|_{C^{1,1}(\mathbb{R}^{n})}\leq 1$. Then $f$ has a definable
extension $F\in C^{1,1}(\mathbb{R}^{n})$ such that
$\|F\|_{C^{1,1}(\mathbb{R}^{n})}\leq C(n)$.

###### Proof.

Combine Theorem 4.5 with Proposition 4.7. ∎

Now we may prove (1.2). Let $F\in C^{1,1}(\mathbb{R}^{n})$ be an extension of
$f$. Then Theorem 4.8 implies that there is a definable extension $G\in
C^{1,1}(\mathbb{R}^{n})$ of $f$ satisfying
$\|G\|_{C^{1,1}(\mathbb{R}^{n})}\leq C(n)\,\|F\|_{C^{1,1}(\mathbb{R}^{n})}$.
Then (1.2) follows easily.

###  4.3. Definable Kirszbraun theorem

As a consequence of the result of [3] used in the proof of Proposition 4.7, a
version of Kirszbraun’s theorem on the extension of Lipschitz functions by an
explicit formula is given in [4, Theorem 1.2]:

_Let $X\subseteq\mathbb{R}^{n}$ be any set and $f:X\to\mathbb{R}^{m}$ a
Lipschitz map with Lipschitz constant $M$, where $\mathbb{R}^{n}$ and
$\mathbb{R}^{m}$ carry the standard inner product (i.e., the Lipschitz
constant $M$ is computed with respect to the Euclidean norms, whence no factor
$\sqrt{n}$). Then_

$\displaystyle F(x)$
$\displaystyle:=\nabla_{\mathbb{R}^{m}}\operatorname{conv}(g)(x,0),\quad
x\in\mathbb{R}^{n},\quad\text{ where}$ $\displaystyle g(x,y)$
$\displaystyle:=\inf_{z\in X}\big{(}\langle
f(z),y\rangle+\frac{M}{2}\|x-z\|^{2}_{2}\big{)}+\frac{M}{2}\|x\|_{2}^{2}+M\|y\|_{2}^{2},\quad(x,y)\in\mathbb{R}^{n}\times\mathbb{R}^{m},$

_defines a Lipschitz extension $F:\mathbb{R}^{n}\to\mathbb{R}^{m}$ of $f$ with
the same Lipschitz constant $M$. _

This follows easily by applying the mentioned theorem of [3] to the $1$-jet
$((0,0),(0,f))$ on
$X\times\\{0\\}\subseteq\mathbb{R}^{n}\times\mathbb{R}^{m}$. Actually, the
result is valid for maps between Hilbert spaces.

As a corollary we get a _definable Kirszbraun theorem_ (since all formulas are
definable):

######  Theorem 4.9.

If $f:X\to\mathbb{R}^{m}$, $X\subseteq\mathbb{R}^{n}$, is a definable
Lipschitz map, then the map $F:\mathbb{R}^{n}\to\mathbb{R}^{m}$ defined above
is a definable Lipschitz extension of $f$ preserving the Lipschitz constant.

The definable Kirszbraun theorem was first proved by Aschenbrenner and Fischer
[1], but the explicit formula gives it immediately.

######  Remark 4.10.

Actually, the definable Kirszbraun theorem [1, Theorem A] holds in any
expansion $\mathfrak{R}=(R,0,1,+,\cdot,<,\ldots)$ of a real closed ordered
field that is _definably complete_ (i.e., each non-empty definable subset of
$R$ which is bounded from above has a least upper bound in $R$). Definable
completeness (which follows from o-minimality) is a necessary condition for
the validity of the definable Kirszbraun theorem; see [1, Proposition 5.2]. A
careful inspection of the proofs of [3, Theorem 3.4] and [4, Theorem 1.2]
shows that the explicit extension formula given above still holds and thus
gives a short alternative proof in this general setting.

###  4.4. Remarks on (1.3)

The obstacle for obtaining (1.3) for all definable moduli of continuity
$\omega$ is that we do not know if a bound of the type (4.6) generally holds.
But, by a result of Pawłucki [15, Theorem 1.2], Whitney jets of class
$C^{m,\omega}$ (not necessarily definable) on a definable closed set
$X\subseteq\mathbb{R}^{n}$ extend to $C^{m,\omega}$-functions and the
extension is by a continuous linear operator which is a finite composite of
operators that either preserve definability or are defined by integration with
respect to a parameter.

Integration may lead out of the o-minimal structure one started with. For
instance, if one starts with globally subanalytic sets and maps, due to a
result of Lion ann Rolin [13], one lands in the algebra of real functions
generated by globally subanalytic functions and their logarithms. For a class
containing globally subanalytic functions and their complex exponentials (thus
oscillatory functions) that is stable under parameterized integration, see
[8]. Generally, for every o-minimal expansion $\mathscr{S}$ of the real field
there is an o-minimal expansion $\widetilde{\mathscr{S}}$ in which the
solutions of Pfaffian equations with $\mathscr{S}$-definable
$C^{1}$-coefficients are $\widetilde{\mathscr{S}}$-definable, by Speissegger
[17]. Thus we may conclude that (1.3) at least holds if the trace norm on the
left-hand side is computed with respect to the functions definable in
$\widetilde{\mathscr{S}}$ instead of those definable in the structure
$\mathscr{S}$ we started with.

In [15], integration with respect to a parameter is used for smoothing
operators that are linear, continuous, and preserve moduli of continuity. In
the proof of the definable Whitney extension theorem of class $C^{m}$ [11, 12,
18] as well as of class $C^{m,\omega}$ [14], smooth cell decomposition and
subtle inequalities for the derivatives of definable functions are used
instead of smoothing operators.

### Acknowledgements

A large part of the work on this paper has been done at Mathematisches
Forschungsinstitut Oberwolfach (_Oberwolfach Research Fellows (OWRF) ID 2244p,
October 31 - November 19, 2022_). We are grateful for the support and the
excellent working conditions.

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