# John property of anisotropic minimal surfaces

Weicong Su and Yi Ru-Ya Zhang  Academy of Mathematics and Systems Science, the
Chinese Academy of Sciences, Beijing 100190, P. R. China<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

For a convex set $K\subset\mathbb{R}^{n}$ and the associated anisotropic
perimeter $P_{K}$, we establish that every $(\epsilon,\,r)$-minimizer for
$P_{K}$ satisfies a local John property. Furthermore, we prove that a certain
class of John domains, including $(\epsilon,\,r)$-minimizers close to $K$,
admits a trace inequality. As a consequence, we provide a more concrete proof
for a crucial step in the quantitative Wulff inequality, thereby complementing
the seminal work of Figalli, Maggi, and Pratelli.

###### Key words and phrases:

John domains, Wulff inequality

###### 2000 Mathematics Subject Classification:

49Q05, 49Q20

Both of the authors are funded by National Key R&D Program of China (Grant No.
2021YFA1003100), the Chinese Academy of Science, and the National Natural
Science Foundation of China (No. 12288201).

##  1\. Introduction

Let $K\subset\mathbb{R}^{n},n\geq 2$ be a convex (open) set such that
$|K|=|B|=\omega_{n}$, where $B$ denotes the standard Euclidean unit ball and
$|\cdot|$ denotes the Lebesgue measure. We denote by $\mathcal{H}^{k}$ the
$k$-dimensional Hausdorff measure. Define

$P_{K}(E)=\int_{\partial^{*}E}f(\nu_{E}(x))\,d\mathcal{H}^{n-1}(x)$

for some positive convex function $f:\mathbb{S}^{n-1}\to\mathbb{R}$. Here, $E$
represents a set of finite perimeter, $\partial^{*}E$ denotes its reduced
boundary, and $\nu_{E}$ refers to the (measure-theoretic) unit outer normal;
see the beginning of Section 2 for more specific definitions.

We usually refer to $P_{K}(E)$ as the _Wulff perimeter_ of $E$, and $K$ as the
_Wulff shape_ corresponding to the surface tension $f$. The (open) set $K$ can
be characterized via $f$ by

$K=\bigcap_{\nu\in\mathbb{S}^{n-1}}\left\\{x\in\mathbb{R}^{n}\colon
x\cdot\nu<f(\nu)\right\\}.$

In particular, when $K=B$, or equivalent $f\equiv 1$, one obtains the standard
Euclidean norm and denotes by

$P(E):=P_{B}(E)=\mathcal{H}^{n-1}(\partial^{*}E)$

the perimeter of $E$ with respect to the Euclidean norm. Analogous to the
standard Euclidean case, we have the Wulff inequality: For any set of finite
perimeter $E\subset\mathbb{R}^{n}$,

$P_{K}(E)\geq n|K|^{\frac{1}{n}}|E|^{\frac{n-1}{n}}.$

Moreover, the equality holds if and only if $E$ is congruent to $K$ up to
under translation and dilation.

In their distinguished work [11], Figalli, Maggi, and Pratelli employed a mass
transportation approach to demonstrate a quantified version of the Wulff
inequality: For every set of finite perimeter $E\subset\mathbb{R}^{n}$ with
$|K|=|E|=|B|$, one has

$P_{K}(E)-P_{K}(K)\geq c(n)\min_{x\in\mathbb{R}^{n}}|E\Delta(x+K)|^{2},$ (1.1)

where the constant is independent of $K$.

An indispensable step in their proof, presented in [11, Theorem 3.4], requires
identifying, for any set $E\subset\mathbb{R}^{n}$ proximate to the Wulff shape
$K$, a corresponding set $F\subset\mathbb{R}^{n}$. This set $F$, while is
close to $E$, supports either a Sobolev-Poincaré inequality of the form:

$\int_{E}\|-Df(x)\|_{*}\,dx\geq
c(n)\inf_{a\in\mathbb{R}}\left(\int_{E}|f(x)-a|^{\frac{n}{n-1}}\,dx\right)^{\frac{n-1}{n}}\quad\text{
for all }\ f\in C_{c}^{1}(\mathbb{R}^{n}),$ (1.2)

wherein one can derive a variant of (1.1) for $F$ with a non-optimal exponent
of $4$ on the right-hand side; or a trace inequality of the form

$\int_{E}\|-Df(x)\|_{*}\,dx\geq c(n)\inf_{a\in\mathbb{R}}\int_{\partial
E}|f(x)-a|\|\nu_{E}(x)\|_{*}\,d\mathcal{H}^{n-1}(x)\quad\text{ for all }\ f\in
C_{c}^{1}(\mathbb{R}^{n}),$ (1.3)

wherein one can precisely establish the inequality (1.1) for $F$ with the
optimal exponent of 2; see [11, Section 1.6] for more discussions. Here,

$\|\nu\|_{*}=\sup_{x\in K}x\cdot\nu\quad\nu\in\mathbb{S}^{n-1},$

denotes the dual norm of the Minkowski norm $\|\cdot\|_{K}$ associated to $K$,
which is defined by

$\|x\|_{K}=\inf\left\\{\lambda>0\colon x\in\lambda K\right\\}.$

Regrettably, their original construction of $F$ is rather abstract, making it
challenging to discern the geometric characteristics of $F$.

On the other hand, a sharp quantitative isoperimetric inequality for the
standard Euclidean norm was proven by Fusco, Maggi, and Pratelli [10] through
the quantification of Steiner symmetrization. Later, Cicalese and Leonardi
provided an alternative argument in [6], based on the regularity of minimal
surfaces. In their approach, they also needed to slightly modify a set
$E\subset\mathbb{R}^{n}$ that is close to the unit ball $B$, in order to
obtain a set $F$ which is $W^{1,\,\infty}$-close to $B$. To achieve this, they
first established a penalized variational problem based on $E$ to obtain the
set $F$, and then apply the regularity results for $(\epsilon,\,r)$-miminzers
(see Definition 1.2 below) to show that $F$ is even $C^{1,\,\alpha}$-close to
$B$ for some $0<\alpha<\frac{1}{2}$. This method is now known as the
_selection principle_ and has found wide application; see, for example, [4,
14, 12].

However, a direct application of the regularity method, namely the selection
principle, in the argument presented in [11] encounters a challenge: the
Lipschitz regularity (uniformly under appropriate normalization) for
$(\epsilon,r)$-minimizers with respect to $P_{K}$ still poses an unresolved
problem when $n\geq 3$; we draw attention to [21] for a related endeavor in
this direction. Indeed, the shape $K$ itself may lack $C^{1,\,\alpha}$
regularity and uniform convexity, thereby complicating prospects for enhanced
regularity. To our knowledge, the most notable advancement in this regard can
be found in [1, Proposition 4.6], where a Lipschitz approximation property was
demonstrated for $(\epsilon,r)$-minimizers of $P_{K}$ associated with a convex
set $K$ of arbitrary form. One may consult the references therein for further
literature on this topic.

###  1.1. Sobolev-Poincaré meets John

Nevertheless, Lipschitz regularity is not fully necessary to support (1.2) (or
(1.3)). Indeed, there have been numerous studies on the geometric
characterizations of domains that support the Sobolev-Poincaré inequality. In
particular, it was shown in [2] and [15] that a John domain admits a
$(p^{*},\,p)$-Sobolev-Poincaré inequality with the same possible exponents
$p^{*}$ as the one for a ball, where $1\leq p<n$ and $p^{*}=\frac{np}{n-p}$.
Later, Buckely and Koskela [5] also proved that the John property is necessary
under some mild geometric assumuptions on the domain. This relation was also
generalized to other metric measure spaces including Carnot groups; see e.g.
[16] for a comprehensive study.

To be specific, let us introduce some notation. Given a Minkowski norm
$\|\cdot\|_{K}$, we define the corresponding distance between two sets
$E,\,F\subset\mathbb{R}^{n}$ by

${\mathop{\mathrm{missing}}{\,dist\,}}_{K}(E,\,F):=\inf\\{\|y-x\|_{K}:x\in
E,y\in F\\}.$

Particularily, when the norm is standard Euclidean one, we simply rewrite the
Euclidean distance between $E,F$ by
${\mathop{\mathrm{missing}}{\,dist\,}}(E,F)$. Moreover, it also gives two
constants

$m_{K}:=\inf\\{\|v\|_{*}:v\in\mathbb{S}^{n-1}\\}\quad\text{and}\quad
M_{K}:=\sup\\{\|v\|_{*}:v\in\mathbb{S}^{n-1}\\}$ (1.4)

where $\|\cdot\|_{*}$ is the dual norm of $\|\cdot\|_{K}$. Then for any
$x\in\mathbb{R}^{n}$,

$\frac{|x|}{M_{K}}\leq\|x\|_{K}\leq\frac{|x|}{m_{K}}.$ (1.5)

As $K$ might not coincide with $-K$, we define

$\eta_{K}=\sup_{x\in\mathbb{S}^{n-1}}\frac{\|x\|_{*}}{\|-x\|_{*}}\geq 1.$
(1.6)

We denote by

$B_{\|\cdot\|_{K},r}(x):=\\{y\in\mathbb{R}^{n}:\|y-x\|_{K}<r\\}$

the ball centered at $x$ with radius $r>0$ with respect to $\|\cdot\|_{K}$.
When $K=B$, we drop the all subindices related to $K$ and, for example, write
$B_{r}(x):=B_{\|\cdot\|_{B},r}(x)$ for brevity. Especially, the standard
Euclidean ball centered at $0$ with radius $r$ is represent as $B_{r}$.

Now we are ready to introduce the concept of John domains.

######  Definition 1.1.

For $J\geq 1$, a (bounded) domain $\Omega\subset\mathbb{R}^{n}$ is said to be
$J$-John with respect to $\|\cdot\|_{K}$ provided that, there exists a
distinguished point $x_{0}\in\Omega$ so that, for any $x\in\Omega$, there
exists a curve $\gamma\subset\Omega$ starting from $x$ ending at $x_{0}$
satisfying the following condition:

$\ell_{\|\cdot\|_{-K}}(\gamma[x,\,y])\leq
J{\mathop{\mathrm{missing}}{\,dist\,}}_{K}(y,\,\partial\Omega)\quad\text{ for
any $y\in\gamma$},$ (1.7)

where $\ell_{\|\cdot\|_{-K}}(\gamma[x,\,y])$ denotes the length of the
subcurve of $\gamma$ joining $x$ to $y$ with respect to the metric induced by
$\|\cdot\|_{-K}.$ We usually call $x_{0}$ the _John center_ of $\Omega$ and
$\gamma$ the _John curve_ joining $x_{0}$ and $x$.

We also need a local version of this definition. Given $s>0$, we say that a
set $\Omega$ is $(J,\,s)$-John if for each $z\in\partial\Omega$ and any $x\in
B_{\|\cdot\|_{K},\,s}(z)\cap\Omega$, there exists a point
$w_{x}\in\Omega\setminus B_{\|\cdot\|_{K},2s}(z)$ with
${\mathop{\mathrm{missing}}{\,dist\,}}_{K}(w_{x},\partial\Omega)\geq 2s$, and
a curve (or even an arc) ${\gamma}\subset\Omega$ joining $x$ to $w_{z}$ so
that such that (1.7) is satisfied for any $y\in\gamma$.

###  1.2. John property of minimizers

As indicated from the preceding discussions, notably the necessity of the John
property in supporting the Sobolev-Poincaré inequality [5], we proceed to show
the possession of the John property by Wulff perimeter minimizers.

Let us first recall the following definition of $(\epsilon,\,r)$-minimizer.

######  Definition 1.2.

Given $\epsilon,\,r>0$, a set of finite perimeter $E$ is an
$(\epsilon,\,r)$-minimizer for $P_{K}$, if for any set
$G\subset\mathbb{R}^{n}$ satisfying $E\Delta G\subset\subset
B_{\|\cdot\|_{K},\,r}(x)$ for some $x\in E$, one has

$P_{K}(E)\leq P_{K}(G)+\epsilon|E\Delta G|.$

Now we are ready to state our main theorem.

######  Theorem 1.3.

Let $\epsilon,\,r>0$ and $E\subset\mathbb{R}^{n}$ be an
$(\epsilon,\,r)$-minimizer for $P_{K}$ with $|K|=|B|$. Then there exists a
constant $J=J\left(n,\,\frac{M_{K}}{m_{K}}\right)>0$ so that each component of
$E$ is a $(J,\,cr)$-John domain for the norm $\|\cdot\|_{K}$ and
$c=c(n,\,\eta_{K})>0$.

To the best of our knowledge, the examination of the John property of
perimeter minimizers was initially undertaken in [8] for global quasiminimal
surfaces for Euclidean perimeters, and subsequently extended to Mumford-Shah
minimizers in [3]. We also refer to [23] for some recent progress on the
properties of John domains related to variational problems.

Our approach to prove Theorem 1.3 relies on a compactness argument together
with the density estimate attributed to each component of a minimizer, a
methodology inspired by the work in [3].

###  1.3. Trace inequality and selection principle

Utilizing Theorem 1.3, we can now leverage the Sobolev-Poincaré inequality
(1.2), as derived in [2], and subsequently employ the selection principle to
deduce the non-optimal exponent of $4$ in (1.1).

Nevertheless, to attain the precise exponent $2$, it is imperative to
establish the validity of (1.3) for functions in the space

$W^{1,\,1}_{K}(\Omega):=\\{u\in L^{1}(\Omega):\|Du\|_{*}\in L^{1}(\Omega)\\}$

within a John domain $\Omega$, subject to supplementary conditions. This is
illustrated in the subsequent proposition.

######  Theorem 1.4.

Let $1\leq\frac{M_{K}}{m_{K}}\leq n$, and $\Omega\subset\mathbb{R}^{n}$ be a
$J$-John domain with respect to $\|\cdot\|_{K}$ and of finite perimeter.
Suppose that $\Omega$ satisfies that, for any $x\in\partial\Omega$ and any
$0<r<r_{0}=c_{0}(n){\mathop{\mathrm{missing}}{\,diam\,}}(\Omega),$ there
exists $\sigma\in(0,\,1)$ so that

${\left|\Omega\cap
B_{\|\cdot\|_{K},\,r}(x)\right|}\leq(1-\sigma){\left|B_{\|\cdot\|_{K},\,r}(x)\right|},$
(1.8)

and

$\mathcal{H}^{n-1}(\partial\Omega\cap B_{\|\cdot\|_{K},r}(x))\leq
C(n)r^{n-1}.$ (1.9)

Then for $\mathcal{H}^{n-1}$-almost every $x\in\partial\Omega$, one has

$\lim_{r\to 0^{+}}{\hbox to0.0pt{\bf\kern
4.49997pt--\hss}\int}\ignorespaces_{\Omega\cap B_{r}(x)}|u(y)-Tu(x)|dy=0$
(1.10)

exists, and we define $\mathcal{H}^{n-1}$-almost everywhere the trace of $u$
on $\partial\Omega$.

Moreover, for any $u\in W^{1,\,1}_{K}(\Omega)$,

$\inf_{c\in\mathbb{R}}\int_{\partial\Omega}|u(x)-c|\|\nu_{\Omega}\|_{*}\,d\mathcal{H}^{n-1}(x)\leq
C(n,\,J)\int_{\Omega}\|Du\|_{*}\,dx.$ (1.11)

Let us expound briefly on how the conditions in Proposition 1.4 are fulfilled
by $(\epsilon,,r)$-minimizers, thereby enabling the application of the
selection principle. As one shall see in the proof of Proposition 1.5,
$\epsilon$ can be chosen depending only on $n$ while $r$ is absolute.

The density estimates presented in Lemma 2.1 below substantiate the assertion
(1.8). Specifically, the upper bound on density (2.5) in Lemma 2.1 inherently
implies (1.9).

Moreover, when the set $E$ is close to $K$ in the sense of (3.3) below, Lemma
3.1 establishes the (global) $J$-John property of $E$. Consequently, we are
empowered to utilize Theorem 1.4 and conclude the following proposition.

######  Proposition 1.5.

Let $1\leq\frac{M_{K}}{m_{K}}\leq n$ and $E_{k}\subset\mathbb{R}^{n}$ be a
sequence of sets of finite perimeter for which $E_{k}$ satisfies (3.3) for
some $\epsilon=\epsilon(n)>0$ small, $|E_{k}|=|K|$, $E_{k}\to K$ in measure
and

$P_{K}(E_{k})-P_{K}(K)\leq 2^{-k}|E_{k}\Delta K|^{2}.$ (1.12)

Then there exists a sequence of $J$-John domains $F_{k}\subset\mathbb{R}^{n}$
so that

$P_{K}(F_{k})-P_{K}(K)\leq\alpha_{k}|F_{k}\Delta K|^{2}$ (1.13)

with $J=J\left(n\right)$ and $\alpha_{k}\to 0$ as $k\to 0$.

Given that the proof of Proposition 1.5 follows a standard approach and
closely mirrors the one found in [13, Pages 560-562], we have included it in
the Appendix for completeness. To some extent, this proposition indicates that
the stability of the (anisotropic) isoperimetric inequality derives from the
stability of the John property of the minimizers under small perturbations.

The structure of our paper unfolds as follows: In Section 2, we begin with
Lemma 2.1, demonstrating that each component of an $(\epsilon,r)$-minimizer
adheres to specific density estimates. This, combined with the compactness
property of $(\epsilon,r)$-minimizers, leads to Theorem 1.3. Subsequently, in
Section 3, we establish the trace inequality presented in Theorem 1.4 and
confirm the selection principle outlined in Proposition 1.5, detailed in the
Appendix.

##  2\. John regularity of $(\epsilon,r)$-minimizer

Let us establish some notation. For a (rectifiable) curve $\gamma$, we denote
by $\ell(\gamma)$ the Euclidean length of $\gamma$. When $\gamma$ is an arc
(i.e. an injective curve), for any pair of points $x,\,y\in\gamma$, denote the
subarc joining $x$ to $y$ as $\gamma[x,\,y]$.

Recall that a measurable set $E\subseteq\mathbb{R}^{n}$ is said to have
_finite perimeter_ if the distributional gradient of its characteristic
function $\mathbf{\chi}_{E}$ is an $\mathbb{R}^{n}$-valued Radon measure
$D\mathbf{\chi}_{E}$ with finite total variation, i.e.,
$|D\mathbf{\chi}_{E}|(\mathbb{R}^{n})<\infty$. According to the Lebesgue-
Besicovitch differentiation theorem for measures, for
$|\mathbf{\chi}_{E}|$-a.e. $x$, it holds

$\lim_{r\to
0^{+}}-\frac{D\mathbf{\chi}_{E}(x+rB^{n})}{|D\mathbf{\chi}_{E}|(x+rB^{n})}=\nu_{E}(x)\quad\text{and}\quad|\nu_{E}(x)|=1.$
(2.1)

The set of points $x$ where this condition holds is called the _reduced
boundary_ of $E$ and is denoted by $\partial^{*}E$. At points on the reduced
boundary, $\nu_{E}(x)$ represents the _measure-theoretic outward unit normal_
to $E$ at $x$. Also, up to changing $E$ in a set of measure zero, one can
assume that $\overline{\partial^{*}E}=\partial E$. We refer the interested
reader to [19, Sections 12 and 15] for more details on sets of finite
perimeter.

Towards Theorem 1.3, we note that, by consider the standard scaling

$E_{x,\,s}=\frac{E-x}{s},$

we have $E_{x,\,s}$ is a $(\epsilon^{\prime},\,r^{\prime})$-minimizer with

$\epsilon^{\prime}=\epsilon s,\quad r^{\prime}=\frac{r}{s}.$

Thus we may normalize our problem by putting an additional assumption that

$\epsilon r\leq 1;$ (2.2)

see e.g.[19, Remark 21.6 & 21.7] for more discussions111Our notion of
$(\epsilon,\,r)$-miminizers is the same as the one in [19]. In particular,
they all belong to the so-called $\omega$-minimal sets; see e.g. [1,
Definition 3.1]. on this.

We start by recalling the relative isoperimetric inequality of Wulff shape:
When $|K|=|B|$, for every set of finite perimeter $E\subset\mathbb{R}^{n}$,
$x\in\mathbb{R}^{n}$ and $r>0$, one has

$P_{K}(E;\,B_{\|\cdot\|_{K},\,r}(x))\geq c(n)\min\left\\{|E\cap
B_{\|\cdot\|_{K},\,r}(x)|,\,|B_{\|\cdot\|_{K},\,r}(x)\setminus
E|\right\\}^{\frac{n-1}{n}}.$ (2.3)

Observe first that the class of convex sets $K\subset\mathbb{R}^{n}$
satisfying

$0\in K,\quad|K|=|B|\quad\text{ and }\quad 1\leq\frac{M_{K}}{m_{k}}\leq n$

forms a compact family. Then the inequality (2.3) is a consequence of the
classical relative isoperimetric inequality in a convex set, where the
constant is independent of $K$ due to the compactness. Then since John’s lemma
[18, Theorem III] yields that, for any convex set $K\subset\mathbb{R}^{n}$,
there exists an affine map $L_{0}$ on $\mathbb{R}^{n}$ so that

$B_{1}\subset L_{0}(K)\subset B_{n},\quad\det L_{0}>0,$

one can apply the trick in [11, Step 2, Proof of Theorem 1.1] to conclude
(2.3) for the general case.

Additionally, for an $(\epsilon,\,r)$-minimizer $E$, we may regard it as a
topologically open set, and equivalently integrate with respect to
$\mathcal{H}^{n-1}$ on either its topological boundary or reduced boundary,
according to [1, Corollary 3.6].

The subsequent density estimates were initially established in [1, Proposition
3.2, 3.4 & 3.5]. However, we require a marginally refined version that is
applicable to each component of a minimizer. The detailed proof is included
due to its significance as a key lemma.

######  Lemma 2.1.

Let $E\subset\mathbb{R}^{n}$ be an $(\epsilon,\,r)$-minimizer for $P_{K}$, and
$E_{0}$ be one of its (connected) component. Then for any $x\in\partial E$,
there exist constants $\sigma=\sigma(n)\in(0,\,1)$ and
$\theta=\theta(n,\,\eta_{K})>1$ so that, whenever $0<s<r$

$\sigma\leq\frac{\left|E_{0}\cap
B_{\|\cdot\|_{K},\,s}(x)\right|}{\left|B_{\|\cdot\|_{K},\,s}(x)\right|}\leq
1-\sigma,$ (2.4)

and

$\theta^{-1}\leq|K|^{-1}s^{1-n}P_{K}(E_{0};\,B_{\|\cdot\|_{K},\,s}(x))\leq\theta,$
(2.5)

where $P_{K}(E_{0};\,B_{\|\cdot\|_{K},\,s}(x))$ denotes the Wulff perimeter of
$E_{0}$ inside (the open set) $B_{\|\cdot\|_{K},\,s}(x)$.

###### Proof.

Suppose for simplicity $n\geq 2$. Fix $x\in\partial E$. As
$|\partial^{*}E|=0$, by coarea formula

$\mathcal{H}^{n-1}(\partial^{*}E\cap\partial B_{\|\cdot\|_{K},s}(x))=0$ (2.6)

for $\mathcal{H}^{1}$-almost every $s\in(0,r)$.

Let $0<s<t<r.$ Define

$F:=(E_{0}\setminus B_{\|\cdot\|_{K},s}(x))\sqcup(E\setminus E_{0}).$

Since $E\Delta F=E_{0}\cap B_{\|\cdot\|_{K},s}(x)\subset\subset
B_{\|\cdot\|_{K},t}(x)$, the $(\epsilon,r)$-minimality of $E$ tells that

$\displaystyle P_{K}(E;B_{\|\cdot\|_{K},t}(x))\leq
P_{K}(F;B_{\|\cdot\|_{K},t}(x))+\epsilon|E\Delta F|.$ (2.7)

Then by Wulff inequality and (2.6),

$\displaystyle|E\Delta F|$ $\displaystyle\leq|E_{0}\cap
B_{\|\cdot\|_{K},s}(x)|^{\frac{n-1}{n}}|B_{\|\cdot\|_{K},s}(x)|^{\frac{1}{n}}\leq\frac{r}{n}P_{K}(E_{0}\cap
B_{\|\cdot\|_{K},s}(x))$
$\displaystyle\leq\frac{r}{n}\left(P_{K}(E_{0};B_{\|\cdot\|_{K},s}(x))+P_{K}(B_{\|\cdot\|_{K},s}(x);E_{0})\right)=:\frac{r}{n}I_{0}.$
(2.8)

Then further recalling (1.5), as $0<s<t$, we have

$\displaystyle I_{0}\leq
P_{K}(E_{0};B_{\|\cdot\|_{K},t}(x))+\int_{E_{0}\cap\partial
B_{\|\cdot\|_{K},s}(x)}\|\nu_{B_{\|\cdot\|_{K},s}(x)}(y)\|_{*}d\mathcal{H}^{n-1}(y)$
$\displaystyle\leq
P_{K}(E_{0};B_{\|\cdot\|_{K},t}(x))+\eta_{K}\int_{E_{0}\cap\partial
B_{\|\cdot\|_{K},s}(x)}\|\nu_{E_{0}\setminus
B_{\|\cdot\|_{K},s}(x)}(y)\|_{*}d\mathcal{H}^{n-1}(y)$ $\displaystyle\leq
P_{K}(E_{0};B_{\|\cdot\|_{K},t}(x))+\eta_{K}P_{K}(E_{0}\setminus
B_{\|\cdot\|_{K},s}(x);B_{\|\cdot\|_{K},t}(x)).$ (2.9)

Now we are ready to show the desired inequalities.

Step 1: The upper bound in (2.5).  Observe the definition of $F$ yields

$P_{K}(F;B_{\|\cdot\|_{K},t}(x))=P_{K}(E_{0}\setminus
B_{\|\cdot\|_{K},s}(x);B_{\|\cdot\|_{K},t}(x))+P_{K}(E\setminus
E_{0};B_{\|\cdot\|_{K},t}(x))$

and

$P_{K}(E;B_{\|\cdot\|_{K},t}(x))=P_{K}(E_{0};B_{\|\cdot\|_{K},t}(x))+P_{K}(E\setminus
E_{0};B_{\|\cdot\|_{K},t}(x)).$

By plugging these two identities together with (2) into (2.7), we arrive at

$\displaystyle P_{K}(E_{0};B_{\|\cdot\|_{K},t}(x))\leq
P_{K}(F;B_{\|\cdot\|_{K},t}(x))+\epsilon|E\Delta F|-P_{K}(E\setminus
E_{0};B_{\|\cdot\|_{K},t}(x))$ $\displaystyle\leq P_{K}(E_{0}\setminus
B_{\|\cdot\|_{K},s}(x);B_{\|\cdot\|_{K},t}(x))+\epsilon|E\Delta F|$
$\displaystyle\leq P_{K}(E_{0}\setminus
B_{\|\cdot\|_{K},s}(x);B_{\|\cdot\|_{K},t}(x))+\frac{\epsilon r}{n}I_{0}.$
(2.10)

As (2) implies

$\displaystyle P_{K}(E_{0}\setminus
B_{\|\cdot\|_{K},s}(x);B_{\|\cdot\|_{K},t}(x))+\frac{\epsilon r}{n}I_{0}$
$\displaystyle\leq\frac{\epsilon
r}{n}P_{K}(E_{0};B_{\|\cdot\|_{K},t}(x))+\left(1+\eta_{K}\frac{\epsilon
r}{n}\right)P_{K}(E_{0}\setminus
B_{\|\cdot\|_{K},s}(x);B_{\|\cdot\|_{K},t}(x)),$ (2.11)

we absorb the term $\frac{\epsilon r}{n}P_{K}(E_{0};B_{\|\cdot\|_{K},t}(x))$
by the left-hand side of (2) and conclude that

$P_{K}(E_{0};B_{\|\cdot\|_{K},t}(x))\leq C(n,\,\eta_{K})P_{K}(E_{0}\setminus
B_{\|\cdot\|_{K},s}(x);B_{\|\cdot\|_{K},t}(x)).$ (2.12)

Now we proceed to estimate $P_{K}(E_{0}\setminus
B_{\|\cdot\|_{K},s}(x);B_{\|\cdot\|_{K},t}(x))$. Observe that,

$\displaystyle P_{K}(E_{0}\setminus
B_{\|\cdot\|_{K},s}(x);B_{\|\cdot\|_{K},t}(x))$
$\displaystyle=\int_{E_{0}\cap\partial
B_{\|\cdot\|_{K},s}(x)}\|\nu_{E\setminus
B_{\|\cdot\|_{K},s}(x)}(y)\|_{*}d\mathcal{H}^{n-1}(y)+P_{K}(E_{0};B_{\|\cdot\|_{K},t}(x)\setminus\overline{B}_{\|\cdot\|_{K},s}(x))$
$\displaystyle=:I_{1}+I_{2}.$ (2.13)

We estimate $I_{1}$ via Wulff inequality

$I_{1}\leq{\eta_{K}}\int_{\partial
B_{\|\cdot\|_{K},s}(x)}\|\nu_{B_{\|\cdot\|_{K},s}(x)}(y)\|_{*}d\mathcal{H}^{n-1}(y)\leq
n|K|\eta_{K}s^{n-1}.$

Then by noticing that $I_{2}\to 0$ as $t\to s^{+}$, we conclude from (2.13)
that

$P_{K}(E_{0}\setminus
B_{\|\cdot\|_{K},s}(x);B_{\|\cdot\|_{K},t}(x))\leq\theta\left(n,\,\eta_{K}\right)s^{n-1}.$

Then the upper bound of $P_{K}(E_{0};B_{\|\cdot\|_{K},t}(x))$ follows from
(2.12).

Step 2: Volume estimates (2.4). Define $m(s)=|E_{0}\cap
B_{\|\cdot\|_{K},s}(x)|$ for $s\in(0,r)$, which is increasing with respect to
$s$. By [1, Remark 2.10], for $\mathcal{H}^{1}$-almost every $s\in(0,r)$,
$m^{\prime}(s)$ exists and satisfies

$m^{\prime}(s)\geq\max\left\\{P_{K}\big{(}E_{0}\cap
B_{\|\cdot\|_{K},s}(x);\,E_{0}\big{)},P_{K}\big{(}E_{0}\setminus
B_{\|\cdot\|_{K},s}(x);\,E_{0}\big{)}\right\\};$ (2.14)

the two terms on the right-hand side are identical when $\eta_{K}=1.$ Observe
that by letting $t\to s^{+}$ in (2) and applying the definition of $I_{0}$ in
(2), we get

$\displaystyle P_{K}(E_{0};\overline{B}_{\|\cdot\|_{k},s}(x))\leq
P_{K}(E_{0}\setminus
B_{\|\cdot\|_{K},s}(x);\overline{B}_{\|\cdot\|_{K},s}(x))+\frac{\epsilon
r}{n}I_{0}$ $\displaystyle\leq P_{K}(E_{0}\setminus
B_{\|\cdot\|_{K},s}(x);\overline{B}_{\|\cdot\|_{K},s}(x))+\frac{\epsilon
r}{n}\left(P_{K}(E_{0};\overline{B}_{\|\cdot\|_{K},s}(x))+P_{K}(\overline{B}_{\|\cdot\|_{K},s}(x);E_{0})\right)$

Combining this inequality with (2.6) and (2.14) one gets that

$\displaystyle\left(1-\frac{\epsilon
r}{n}\right)P_{K}(E_{0};\overline{B}_{\|\cdot\|_{k},s}(x))\leq
P_{K}(E_{0}\setminus
B_{\|\cdot\|_{K},s}(x);\overline{B}_{\|\cdot\|_{K},s}(x))+\frac{\epsilon
r}{n}P_{K}(\overline{B}_{\|\cdot\|_{K},s}(x);E_{0})$
$\displaystyle=P_{K}(E\setminus B_{\|\cdot\|_{K},s}(x);\,E_{0})+\frac{\epsilon
r}{n}P_{K}\big{(}E_{0}\cap
B_{\|\cdot\|_{K},s}(x);\,E_{0}\big{)}\leq\left(1+\frac{\epsilon
r}{n}\right)m^{\prime}(s).$ (2.15)

Then since

$I_{0}=P_{K}(E_{0};B_{\|\cdot\|_{K},s}(x))+P_{K}(B_{\|\cdot\|_{K},s}(x);E_{0}),$

(2.14) and (2) together with the assumption (2.2) and $n\geq 2$ give that

$I_{0}\leq\frac{1+\frac{\epsilon r}{n}}{1-\frac{\epsilon
r}{n}}m^{\prime}(s)+m^{\prime}(s)\leq 4m^{\prime}(s).$

Hence, we further apply isoperimetric inequality to $E_{0}\cap
B_{\|\cdot\|_{K},s}(x)$ together with the estimate

$P_{K}(E\cap B_{\|\cdot\|_{K},s}(x))\leq I_{0}$

from (2) to get

$\displaystyle n(m(s))^{\frac{n-1}{n}}|K|$ $\displaystyle\leq P_{K}(E\cap
B_{\|\cdot\|_{K},s}(x))\leq I_{0}\leq 4m^{\prime}(s),$ (2.16)

from which we deduce $m(s)\geq\left(|K|/4\right)^{n}s^{n}$ by integration over
$s$. This concludes the lower bound in (2.4).

Moreover, [1, Proposition 3.4] has proven an $(\epsilon,r)$-minimizer $E$
satisfies

$|E\cap B_{\|\cdot\|_{K},s}(x)|\leq(1-\sigma)|B_{\|\cdot\|_{K},s}(x)|$

for some $\sigma=\sigma(n)\in(0,1)$ and $s\in(0,r)$. As a result, the upper
bound in (2.4) follows as $m(s)\leq|E\cap B_{\|\cdot\|_{K},s}(x)|$. This
completes the proof of (2.4).

Step 3: The lower bound in (2.5).  It remains to show the lower bound in
(2.5), which is a direct consequence of (2.3) and (2.4). Thus, (2.5) holds,
and we conclude the lemma. ∎

With the help of the density estimates, we prove the following _plumpness_ for
the $(\epsilon,\,r)$-minimizers, a terminology used in e.g. [26, 2.2]. As each
component of an $(\epsilon,\,r)$-minimizer is also an
$(\epsilon,\,r)$-minimizer, we may only consider connected ones for the time
being.

######  Lemma 2.2.

Let $E$ be a connected $(\epsilon,\,r)$-minimizer for $P_{K}$ and
$x\in\partial E$. Then there exists $0<c_{0}=c_{0}(n,\,\eta_{K})<1$ and a ball
$D$ with radius $c_{0}\rho$ and $0<\rho<r$ so that,

$D\subset B_{\|\cdot\|_{K},\,\rho}(x)\cap E.$

###### Proof.

Suppose that the claim of the lemma fails for some $c>0$ to be determined.
Then for any $y\in E\cap B_{\|\cdot\|_{K},\,\rho}(x)$, the ball
$B_{\|\cdot\|_{K},\,c\rho}(y)$ must intersect $\partial E$. In particular, by
Lemma 2.1, we have

$\theta^{-1}(c\rho)^{n-1}\leq P_{K}(E;\,B_{\|\cdot\|_{K},\,2c\rho}(y)).$
(2.17)

Moreover, Lemma 2.1 also yields that, for $\sigma=\sigma(n)\in(0,\,1)$,

$|E\cap B_{\|\cdot\|_{K},\,\rho}(x)|\geq\sigma\omega_{n}\rho^{n},$ (2.18)

and for any $y\in E\cap B_{\|\cdot\|_{K},\,\rho}(x),$

$|E\cap B_{\|\cdot\|_{K},\,2c\rho}(y)|\leq\omega_{n}(2c\rho)^{n}.$ (2.19)

In addition, by applying Bescovitch covering theorem [9, Section 1.5.2] to

$\\{B_{\|\cdot\|_{K},\,2c\rho}(y)\\}_{y\in E\cap
B_{\|\cdot\|_{K},\,\rho}(x)},$

we conclude that, there exists $N$-many $y_{k}$’s with
$N\in\mathbb{N}\cup\\{\infty\\}$ and $y_{k}\in E$ so that, for any $z\in E$

$\chi_{E\cap B_{\|\cdot\|_{K},\,\rho}(x)}(z)\leq\sum_{k}\chi_{E\cap
B_{\|\cdot\|_{K},\,2c\rho}(y_{k})}(z)\leq C(n).$ (2.20)

Then according to (2.18) and (2.19), we have

$\sigma\omega_{n}\rho^{n}\leq|E\cap
B_{\|\cdot\|_{K},\,\rho}(x)|\leq\sum_{k}|E\cap
B_{\|\cdot\|_{K},\,2c\rho}(y_{k})|\leq N\omega_{n}(2c\rho)^{n},$

which implies $N\geq c(n)c^{-n}$.

On the other hand, (2.17) applied to each $B_{\|\cdot\|_{K},\,2c\rho}(y_{k})$
together with Lemma 2.1 and (2.20) yields

$\theta^{-1}(c\rho)^{n-1}c(n)c^{-n}\leq\sum_{k}P_{K}(E;\,B_{\|\cdot\|_{K},\,2c\rho}(y_{k}))\leq
C(n)P_{K}(E;\,B_{\|\cdot\|_{K},\,3\rho}(x))\leq C(n)\theta\rho^{n-1}.$

Thus by choosing $0<c\leq c_{0}=c_{0}\left(n,\,\eta_{K}\right)$ small so that

$c_{0}^{-1}\geq C(n)\theta^{2},$

we obtain the desired contradiction and conclude the lemma. ∎

Also recall that $(\epsilon,\,r)$-minimizers are precompact.

######  Lemma 2.3 ([1, Proposition 3.3],[19, Theorem 21.14]).

Let $\\{E_{k}\\}$ be a sequence of $(\epsilon,\,r)$-minimizers in
$\mathbb{R}^{n}$. Then up to picking a subsequence and relabeling, $E_{k}$
converge locally in measure to $E$, and $E$ is also an
$(\epsilon,\,r)$-minimizer. Moreover, both

$D{\chi_{E_{k}}}\rightharpoonup D{\chi_{E}}\quad\text{ and
}\quad\|D{\chi_{E_{k}}}\|_{K}\rightharpoonup\|D{\chi_{E}}\|_{K}$

weakly in measure.

Further assume $E_{k}$ are equibounded. Then we have

$\partial E_{k}\to\partial E\quad\text{ in }(\mathcal{C}^{n},d_{H})\quad\text{
as }k\to+\infty,$

where $\mathcal{C}^{n}$ is the space consisting of all nonempty compact sets
in $\mathbb{R}^{n}$ equipped with the Hausdorff metric $d_{H}$.

Now we show the following lemma, which is in the spirit similar to that of [7,
Lemma 68.14]; see also [3, Section 19].

######  Lemma 2.4.

Let $E$ be an $(\epsilon,\,r)$-minimizer and $z_{0}\in E$. Write

$d_{0}={\mathop{\mathrm{missing}}{\,dist\,}}_{K}(z_{0},\,\partial E).$

Suppose that $d_{0}\ll r.$ Then there exists
$C_{1}=C_{1}\left(n,\frac{M_{K}}{m_{K}}\right)\geq 1$ and a curve
$\gamma\subset E\cap B_{\|\cdot\|_{K},\,C_{1}d_{0}}(z_{0})$ so that

${\mathop{\mathrm{missing}}{\,dist\,}}_{K}(\gamma,\,\partial E)\geq
C_{1}^{-1}d_{0}$ (2.21)

and $\gamma$ goes from $z_{0}$ to some point $z_{1}\in E$ such that

$d_{1}:={\mathop{\mathrm{missing}}{\,dist\,}}_{K}(z_{1},\,\partial E)\geq
2d_{0}.$

###### Proof.

First of all, we observe that, for a class of convex sets $K$ with the same
volume and with $\frac{M_{K}}{m_{K}}$ uniformly bounded from above, all the
metrics induced by $\|\cdot\|_{K}$ are $\frac{M_{K}}{m_{K}}$-bi-Lipschitz
equivalent. Thus we may fix one of such a convex set $K$.

We proceed by contradiction, via the compactness given by Lemma 2.3. Suppose
that there exists a sequence of $(\epsilon,\,r)$-minimizers $E_{k}$ with
$z_{0,\,k}\in E_{k}$ so that (2.21) fails for $C_{1}=2^{k}$ and

$d_{0,\,k}:={\mathop{\mathrm{missing}}{\,dist\,}}_{K}(z_{0,\,k},\,\partial
E_{k})\to 0.$

Then by considering

$F_{k}=\frac{E_{k}-x_{k}}{d_{0,\,k}},$

where

$\|x_{k}-z_{0,k}\|_{K}={\mathop{\mathrm{missing}}{\,dist\,}}_{K}(z_{0,k},\partial
E_{k}),$

we have the ball $B_{\|\cdot\|_{K},1}((z_{0,k}-x_{k})/d_{0,k})\subset F_{k}$
and our lemma fails for every curve starting at $z_{0,k}$ with $C_{1}=2^{k}$.

Up to passing to a subsequence, $F_{k}$ converge to $F$ locally in measure
according to Lemma 2.3 with $B_{\|\cdot\|_{K},1}(x)\subset F$ for some
$x\in\mathbb{R}^{n}$. Let $F_{0}$ be the component of $F$ containing
$B_{\|\cdot\|_{K},1}(x)$. Then Lemma 2.2 (applied to $F_{0}$) implies tha
there exists a ball, say $B_{\|\cdot\|_{K},\,3}(y_{1})$, inside $F_{0}$. As
connected open sets are path-connected, we can join $x$ to $y_{1}$ by a curve
$\gamma\subset F_{0}$, and set

$\delta:={\mathop{\mathrm{missing}}{\,dist\,}}_{K}({\gamma},\,\partial
F_{0})>0.$

Note that, by the lower density estimates in Lemma 2.1, the limit of
components of $B_{\|\cdot\|_{K},\,R}\setminus F_{k}$ does not vanish for any
$R>0$. Then by choosing $k$ large enough, Lemma 2.3 tells that

${\mathop{\mathrm{missing}}{\,dist\,}}_{K}(\gamma,\,\partial
F_{k})\geq\frac{\delta}{2},$

together with ${\mathop{\mathrm{missing}}{\,dist\,}}_{K}(y_{1},\,\partial
F_{k})\geq 2.$ Thus $\gamma$ satisfies (2.21) with

$C_{1}=\max\left\\{{\mathop{\mathrm{missing}}{\,diam\,}}_{K}(\gamma),\,\frac{2}{\delta}\right\\}>0.$

This gives the desired contradiction and conclude the lemma. ∎

######  Remark 2.5.

For the curve $\gamma\subset B_{\|\cdot\|_{K},\,C_{1}d_{0}}(z_{0})$ satisfying
the assumption in Lemma 2.4, we may replace $\gamma$ by a rectifiable curve
$\widetilde{\gamma}$ joining $z_{0}$ to $z_{1}$, so that

$\ell_{\|\cdot\|_{-K}}(\widetilde{\gamma})\leq
C_{2}{\mathop{\mathrm{missing}}{\,dist\,}}_{K}(\widetilde{\gamma},\,\partial
F_{k})\quad\text{and}\quad{\mathop{\mathrm{missing}}{\,dist\,}}_{K}(\widetilde{\gamma},\,\partial
F_{k})\geq C_{2}^{-1}d_{0},$ (2.22)

with $C_{2}=C_{2}\left(n,\,C_{1},\,\frac{M_{K}}{m_{K}}\right)$. This has been
proven in [20, Lemma 2.6] for the Euclidean norm, and we supply the proof in
the case of anisotropic norms.

Indeed, via the same idea used in [20, Lemma 2.6], one is able to find a
rectifiable curve $\widetilde{\gamma}$ joining $z_{0}$ to $z_{1}$, such that

$\displaystyle car_{l}(\widetilde{\gamma},C_{3})$
$\displaystyle:=\bigcup\big{\\{}B_{\|\cdot\|_{K},\ell_{\|\cdot\|_{-K}}(\widetilde{\gamma}[z_{0},y])/C_{3}}(y):y\in\widetilde{\gamma}\setminus\\{z_{0}\\}\big{\\}}$
$\displaystyle\subset\bigcup\big{\\{}B_{\|\cdot\|_{K},{\mathop{\mathrm{missing}}{\,dist\,}}_{K}(\gamma[z_{0},y])/C_{1}^{2}}(y):y\in\gamma\setminus\\{z_{0}\\}\big{\\}}=:car_{d}(\gamma,C_{1}^{2}),$
(2.23)

with $C_{3}=C_{3}\left(n,C_{1},\frac{M_{K}}{m_{K}}\right)$. As
$car_{d}(\gamma,C_{1}^{2})\subset E$ follows from Lemma 2.4, (2.5) implies

$car_{l}(\widetilde{\gamma},C_{3})\subset E.$ (2.24)

Observe that for each point $y\in\widetilde{\gamma}$ with $\|y-z_{0}\|_{K}\leq
d_{0}/2$, ${\mathop{\mathrm{missing}}{\,dist\,}}_{K}(y,\partial\Omega)\geq
d_{0}/2$ holds by using triangle inequality. In addition, for each point
$y\in\widetilde{\gamma}$ with $\|y-z_{0}\|_{K}>d_{0}/2$, via (2.24) and (1.5)
we deduce

${\mathop{\mathrm{missing}}{\,dist\,}}_{K}(y,\partial
E)\geq\frac{\ell_{\|\cdot\|_{-K}}(\widetilde{\gamma}[z_{0},y])}{C_{3}}\geq\frac{\|y-z_{0}\|_{-K}}{C_{3}}\geq\frac{m_{K}\|y-z_{0}\|_{K}}{C_{3}M_{K}}\geq\frac{m_{K}d_{0}}{2C_{3}M_{K}}.$

Then it follows that
${\mathop{\mathrm{missing}}{\,dist\,}}_{K}(\widetilde{\gamma},\partial
E)\geq\frac{m_{K}d_{0}}{2C_{3}M_{K}}$. Further observe that $z_{1}\in
B_{\|\cdot\|_{K},C_{1}d_{0}}(z_{0})$. This implies by applying triangle
inequality and (1.5)

${\mathop{\mathrm{missing}}{\,dist\,}}_{K}(z_{1},\partial
E)\leq\|z_{0}-z_{1}\|_{K}+{\mathop{\mathrm{missing}}{\,dist\,}}_{K}(z_{0},\partial
E)\leq\frac{M_{K}}{m_{K}}\|z_{1}-z_{0}\|_{K}+d_{0}\leq\frac{M_{K}}{m_{K}}(C_{1}+1)d_{0}.$

This estimate, coupled with (2.24), gives (2.22) by letting
$C_{2}=\frac{2M_{K}^{2}}{m_{K}^{2}}(C_{1}+1)C_{3}^{2}$.

Now we are ready to prove Theorem 1.3.

###### Proof of Theorem 1.3.

Let $z_{0}\in E$ be a point close to $\partial E$, and we construct the
desired curve $\gamma\subset E$ starting from $z_{0}$ by induction.

First by Lemma 2.4 (together with Remark 2.5), there exists $z_{1}\in E$ with

$d_{1}:={\mathop{\mathrm{missing}}{\,dist\,}}_{K}(z_{1},\,\partial E)\geq
2{\mathop{\mathrm{missing}}{\,dist\,}}_{K}(z_{0},\,\partial E):=2d_{0}$

so that one can construct a (smooth) curve $\gamma_{0}\subset E$ joining
$z_{0}$ to $z_{1}$ and satisfies (2.22) with constant $C_{2}$ and
$\ell_{\|\cdot\|_{-K}}({\gamma}_{0})\leq C_{2}d_{0}$.

Now we continue our construction by setting up a curve $\gamma_{1}$ starting
at $z_{1}$ and ending at some point $z_{2}\in E$ so that

$d_{2}:={\mathop{\mathrm{missing}}{\,dist\,}}_{K}(z_{2},\,\partial E)\geq
2d_{1}.$

Again by Lemma 2.4 (together with Remark 2.5), we have that

$\gamma_{1}\subset
B_{\|\cdot\|_{K},\,C_{2}d_{1}}(z_{1}),\quad{\mathop{\mathrm{missing}}{\,dist\,}}_{K}(\gamma_{1},\,\partial
E)\geq(C_{2})^{-1}d_{1},\quad\text{ and
}\quad\ell_{\|\cdot\|_{-K}}({\gamma}_{1})\leq C_{2}d_{1}.$

We iterate the construction until the process cannot continue, i.e. the
density estimates Lemma 2.1 fails outside a ball of radius $c(n)r$. Now by
concatenating these curves, letting $\gamma=\cup_{k}\gamma_{k}$ and
reparametrizing it via arc-length, our theorem follows. ∎

##  3\. Trace inequality in John domains

According to the normalization given by John’s lemma [18, Theorem III], there
exists $r=r(|K|)>0$ and an affine map $L$ of $\mathbb{R}^{n}$ so that

$B_{r}\subset L(K)\subset B_{nr}$ (3.1)

with $\det L=1.$ As $|K|=|B|$, it follows that

$r<1,\quad m_{K}\leq 1\leq
M_{K}\quad\text{and}\quad\eta_{K}\leq\frac{M_{K}}{m_{K}}\leq n.$ (3.2)

We always assume that $K$ satisfies this normalization condition in this
section.

By [11], towards the stability of Wulff inequalities with respect to $P_{K}$,
it suffices to consider the case where the set $E\subset\mathbb{R}^{n}$
satisfies

$(1-\delta)K\subset E\subset(1+\delta)K$ (3.3)

for some $\delta=\delta(n)>0$. When $\delta$ is small enough, we first show
that any $(\epsilon,\,r)$-minimizer is a John domain based on Theorem 1.3.

######  Lemma 3.1.

Let $0<\delta\leq\frac{c}{3n}r$, where $c$ is the constant in Theorem 1.3.
Then any $(\epsilon,\,r)$-minimizer $E$ satisfies (3.3) is a $J$-John domain
with $J=J(n).$

######  Remark 3.2.

Since $\eta_{K}\leq\frac{M_{K}}{m_{K}}\leq n$ by the additional assumption
(3.2), one has $c=c(n)>0$. Moreover, in the proof of Proposition 1.5 one
chooses $r>0$ to be absolute, and then $\delta$ depends only on $n$
eventually.

###### Proof of Lemma 3.1.

We show that every point $z\in E$ can be joined to the John center $0$ of $E$
which is also the origin, by a John curve $\gamma_{z}\subset E$ so that for
any $a\in\gamma_{z}[z,0]$,

$\ell_{\|\cdot\|_{-K}}(\gamma_{z}[z,\,a])\leq
J{\mathop{\mathrm{missing}}{\,dist\,}}_{K}(a,\,\partial E)$ (3.4)

holds for some $J=J(n)$.

Towards this, for given $z\in E$, let $y\in\partial K$ be the unit vector so
that $z=\lambda y$ for some $\lambda>0$. Choose the outermost point
$z_{0}\in\partial E$ which is in the same direction of $y$, i.e.
$z_{0}:=\lambda_{0}y$, and which is in the same component of $E$ as $z$. Then
$\lambda_{0}>\lambda$.

To achieve our proof, we discuss in two cases.

Case 1: $z\in(1-2\delta)K$. Observe that $z_{0}$, $z$ and $0$ are on the same
line. Thus, the convexity of $(1-2\delta)K$ and the assumption (3.3) ensure
that the line segment $L_{z}\subset E$ joining $z$ to $0$ is the desired John
curve with John constant $1$.

Case 2: $z\in E\setminus(1-2\delta)K$. We aim to find a curve $\beta_{z}$
joining $z$ to a point $z_{1}\in(1-2\delta)K$. Since we can join $z_{1}$ to
the origin via a line segment according to Case 1, the desired John curve is
obtained by concatenating these two curves

To this end, (3.3) implies $\lambda\in[1-2\delta,1+\delta)$ and
$\lambda_{0}\in(\lambda,1+\delta)$ so that

$\|z_{0}-z\|_{K}=(\lambda_{0}-\lambda)\|y\|_{K}=\lambda_{0}-\lambda\leq
3\delta.$

This implies $z\in B_{\|\cdot\|_{K},3\eta_{K}\delta}(z_{0})$. As $\eta_{K}\leq
n$ by (3.2), $3\eta_{K}\delta\leq 3n\delta\leq cr$.

On the other hand, as Theorem 1.3, together with (3.2), implies $E$ is a
$(J_{0},\ cr)$-John domain for $J_{0}=J_{0}(n)$, we can join $z$ to some point
$z_{1}\in E\setminus B_{\|\cdot\|_{K},2cr}(z_{0})$ by a curve
$\beta_{z}\subset E$ satisfying

${\mathop{\mathrm{missing}}{\,dist\,}}_{K}(z_{1},\partial E)=2cr\ \text{ and
}\ \ell_{\|\cdot\|_{-K}}(\beta_{z}[z,\,a])\leq
J_{0}{\mathop{\mathrm{missing}}{\,dist\,}}_{K}(a,\,\partial\Omega)\quad\text{for
any }a\in\beta_{z}[z,\,z_{1}].$ (3.5)

Since $2cr\geq 3\eta_{K}\delta$, and $\partial
E\subset(1+\delta)K\setminus(1-\delta)K,$ we conclude that
$z_{1}\in(1-2\delta)K$. Now by joining $z_{1}$ to the origin via a line
segment $L_{z_{1}}\subset E$ via Case 1, we further get that

$\ell_{\|\cdot\|_{-K}}(L_{z_{1}}[z_{1},\,a])\leq{\mathop{\mathrm{missing}}{\,dist\,}}_{K}(a,\,\partial
E)\qquad\text{for any }a\in L_{z_{1}}[z_{1},\,0].$ (3.6)

Set $\gamma_{z}:=\beta_{z}\cup L_{z_{1}}$, which is a curve joining $z$ to
$0$. Now we show that $\gamma_{z}$ is the desired John curve. By (3.5), it
suffices to check points $b\in L_{z_{1}}.$

Since $b\in L_{z_{1}}$ with $L_{z_{1}}$ the segment joining $z_{1}$ to $0$,
then by the triangle inequality, the assumption that $\partial
E\subset(1+\delta)K\setminus(1-\delta)K,$ together with the fact that
${\mathop{\mathrm{missing}}{\,dist\,}}_{K}(z_{1},\,\partial
E)>3\eta_{K}\delta$,

$\displaystyle\frac{1}{3}{\mathop{\mathrm{missing}}{\,dist\,}}_{K}(z_{1},\,\partial
E)\leq$
$\displaystyle{\mathop{\mathrm{missing}}{\,dist\,}}_{K}(z_{1},\,\partial
E)-2\eta_{K}\delta\leq{\mathop{\mathrm{missing}}{\,dist\,}}_{K}(z_{1},\,\partial((1-\delta)K)$
$\displaystyle\leq$
$\displaystyle{\mathop{\mathrm{missing}}{\,dist\,}}_{K}(b,\,\partial((1-\delta)K))\leq{\mathop{\mathrm{missing}}{\,dist\,}}_{K}(b,\,\partial
E).$

Therefore, by applying (3.5) with $a=z_{1}$, (3.6) with $a=b$, the
construction of $\gamma_{z}$ tells

$\displaystyle\ell_{\|\cdot\|_{-K}}(\gamma_{z}[z,b])=$
$\displaystyle\ell_{\|\cdot\|_{-K}}(\beta_{z}[z,\,z_{1}])+\ell_{\|\cdot\|_{-K}}(L_{z_{1}}[z_{1},\,b])$
$\displaystyle\leq$ $\displaystyle
J_{0}{\mathop{\mathrm{missing}}{\,dist\,}}_{K}(z_{1},\,\partial
E)+{\mathop{\mathrm{missing}}{\,dist\,}}_{K}(b,\,\partial
E)\leq(3J_{0}+1){\mathop{\mathrm{missing}}{\,dist\,}}_{K}(b,\,\partial E).$

This imples (3.4) when $b\in L_{z_{1}}$. This completes the proof. ∎

We also record the following observation.

######  Lemma 3.3.

Let $\Omega\subset\mathbb{R}^{n}$ be a John domain with John center
$x_{0}\in\Omega$. Then for any $x\in\partial\Omega$, there exists (at least)
one curve $\gamma_{x}\subset\overline{\Omega}$ with
${\gamma}_{x}\setminus\\{x\\}\subset\Omega$ from $x$ to $x_{0}$ for which
(1.7) holds.

###### Proof.

Take a sequence of points $x_{i}\in\Omega$ approaching $x$, and the
corresponding John curve ${\gamma}_{x_{i}}\subset\Omega$ joining $x$ to
$x_{0}$. Then $\ell(\beta_{x_{i}})$ is uniformly bounded according to (1.7).
Now by parametrizing via arc length and up to relabeling the sequence, Arzelá-
Ascoli lemma yields the uniform convergence of ${\gamma}_{i}$ to some curve
${\gamma}_{x}\subset\overline{\Omega}$ joining $x$ to $x_{0}$.

Moreover, the uniform convergence also implies that ${\gamma}_{x}$ satisfies
(1.7). This gives

${\gamma}_{x}\setminus\\{x\\}\subset\Omega.$

This concludes the lemma. ∎

Now we proceed to show the main result of this section. For a Borel set $E$
and $x\in E$, we denote by $E^{(\lambda)}$ the set

$E^{(\lambda)}:=\left\\{x\in\mathbb{R}^{n}:\lim_{r\to 0}\frac{|E\cap
B_{r}(x)|}{|B_{r}(x)|}=\lambda\right\\}.$

Then the _measure-theoretic boundary_ $\partial_{*}E$ of $E$ is set defined as
$\partial_{*}E:=\mathbb{R}^{n}\setminus(E^{(0)}\cup E^{(1)})$. Now we
introduce the definition of admissible domain.

######  Definition 3.4.

A bounded domain $\Omega\subset\mathbb{R}^{n}$ with finite perimeter is said
to be admissible provided

  1. (i)

The measure-theoretic boundary of $\Omega$ almost coincides with its
topological boundary, i.e.

$\mathcal{H}^{n-1}(\partial\Omega\setminus\partial_{*}\Omega)=0.$ (3.7)

  2. (ii)

For any $x\in\partial\Omega$, there exists a positive constant
$\Theta=\Theta(\Omega)$ and a ball $B_{r}(x)$ so that

$\mathcal{H}^{n-1}((\partial_{*}\Omega)\cap(\partial_{*}E))\leq\Theta\mathcal{H}^{n-1}(\Omega\cap(\partial_{*}E))$
(3.8)

holds for each measurable set $E\subset\overline{\Omega}\cap B_{r}(x)$.

In [27], it is shown that for an admissible domain
$\Omega\subset\mathbb{R}^{n}$, the trace of the function $u\in
W^{1,1}(\Omega)$ can be defined in $L^{1}(\partial\Omega)$. To be more
precise, as long as we prove that all the John domains
$\Omega\subset\mathbb{R}^{n}$ satisfying the assumptions in Proposition 1.4
are the admissible domains, [27, Theorem 5.14.4] immediately ensure the
existence of the trace for each $u\in W^{1,1}(\Omega)$. Additionally, [27,
Theorem 5.11.1] gives

$\inf_{c}\int_{\partial\Omega}|u(x)-c|\,d\mathcal{H}^{n-1}(x)\leq
C(\Omega)\int_{\Omega}|Du(x)|\,dx.$

However, the inequality above is insufficient for our purposes, even in the
Euclidean case, as the dependence of $C(\Omega)$ on $\Omega$ is not explicitly
defined. To address this issue, we present an alternative proof based on the
John property, which provides an explicit dependence on the parameters.

Let us introduce the Whintey decomposition for an open set
$\Omega\subsetneqq\mathbb{R}^{n}$. For a constant $c>0$ and any cube
$Q\subset\mathbb{R}^{n}$ with center $x_{Q}=(a_{1},\cdots,a_{n})$ and sides
parallel to the coordinate axis, we let $\ell(Q)$ be the edge length of $Q$
and then rewrite

$\\{x=(x_{1},\cdots,x_{n})\in\mathbb{R}^{n}:-c\ell(Q)/2\leq x_{i}-a_{i}\leq
c\ell(Q)/2\\}$

as $cQ$ for brevity. Now we can state the following lemma on Whitney
decomposition, see e.g. [22, Chapter VI].

######  Lemma 3.5 (Whitney decomposition).

For any open set $\Omega\neq\mathbb{R}^{n}$ there exists a collection
$\mathscr{F}=\\{Q_{j}\\}_{j\in\mathbb{N}}$ of countably many closed dyadic
cubes such that

$\bigcup_{Q\in\mathscr{F}}Q=\Omega,\quad\chi_{\Omega}\leq\sum_{Q\in\mathscr{F}}\chi_{\frac{11}{10}Q}\leq
C(n)\chi_{\Omega}.$ (3.9)

Moreover, for any $Q_{i},\,Q_{j}\in\mathscr{F}$ with $Q_{i}\cap
Q_{j}\neq\emptyset$, one has

$\sqrt{n}\ell(Q_{i})\leq{\mathop{\mathrm{missing}}{\,dist\,}}(Q_{i},\partial\Omega)\leq
4\sqrt{n}\ell(Q_{i})\quad\text{and}\quad\frac{1}{4}\leq\frac{\ell(Q_{i})}{\ell(Q_{j})}\leq
4.$ (3.10)

We first show that every John domain in Theorem 1.4 is admissible.

######  Lemma 3.6.

A bounded $J$-John domain $\Omega\subset\mathbb{R}^{n}$ with finite perimeter
is admissible, provided (1.8) and (1.9) are satisfied. Then every function
$u\in W_{K}^{1,1}(\Omega)$ has a trace in $L^{1}(\partial\Omega)$.

###### Proof.

Since $\Omega$ is John, then for any $x\in\partial\Omega$ and $0<r<r_{0}$,
$\Omega\cap B_{\|\cdot\|_{K},\,r}(x)$ contains a ball
$B_{\|\cdot\|_{K},c_{0}r}$ for some $y\in\Omega\cap B_{\|\cdot\|_{K},\,r}(x)$
and $c_{0}=c_{0}(n,\,J)>0$. Therefore we have

${\left|\Omega\cap B_{\|\cdot\|_{K},\,r}(x)\right|}\geq
c(n,\,J){\left|B_{\|\cdot\|_{K},\,r}(x)\right|}.$

This, combining with (3.7), implies $x\in\partial_{*}\Omega$. Thus
$\partial\Omega=\partial_{*}\Omega$ and, in particular, (3.7) is satisfied.

Suppose that $x_{0}\in\Omega$ is the John center of $\Omega$ and $\mathscr{F}$
is the Whitney decomposition of $\Omega$. Now we verify (3.8). Toward this,
for any $x\in\partial\Omega$, we first choose $r>0$ with $x_{0}\notin
B_{r}(x)$, and let $E\subset\overline{\Omega}\cap B_{r}(x)$.

According to (3.8), we may assume that

$\mathcal{H}^{n-1}(\Omega\cap\partial_{*}E)<\infty.$

Then since $\Omega$ is of finite perimeter, Federer’s theorem [19, Theorem
16.2] together with the fact

$\partial_{*}E=(\partial_{*}E\cap\partial\Omega)\cup(\Omega\cap\partial_{*}E)$

implies that $E$ is of finite perimeter. Consequently, we may further assume
that $E$ has finite perimeter.

Step 1: A regularization of $\partial E$.  We replace $E$ by a more regular
set so that it has certain lower density estimate.

To be more specific, we first consider a minimizer $U\subset\mathbb{R}^{n}$ of
the Plateau problem:

$\inf\\{P(V;\Omega):V\cap\partial\Omega=\partial
E\cap\partial\Omega,\,V\cap\Omega\subset\Omega\cap B_{r}(x)\\},$

From [19, Proposition 12.29] and [19, Example 16.13] it follows that ${U}$
exists and is a local perimeter minimizers in $\Omega\cap B_{r}(x)$.

Moreover,

$c(n)\rho^{n-1}\leq P(U;B_{\rho}(y))\leq n|B|\rho^{n-1}\quad\text{ for any
$B_{\rho}(y)\subset\subset\Omega$ with $y\in\Omega\cap\partial U$},$ (3.11)

is satisfied by [19, Theorem 16.14]. In particular, combining (3.10) and
(3.11), for any Whitney cube $Q\in\mathscr{F}$, there exists a constant
$c_{1}=c_{1}(n)$, such that

$\mathcal{H}^{n-1}\left(\partial U\cap\frac{11}{10}Q\right)\geq
c_{1}(n)l(Q)^{n-1}.$ (3.12)

Also, (3.11), the definition of $\partial_{*}U$ and Federer’s theorem [19,
Theorem 16.2] imply

$\partial U\cap\Omega=\partial_{*}U\cap\Omega\ \text{ and }\
\mathcal{H}^{n-1}(\Omega\cap(\partial U\setminus\partial^{*}U))=0,$ (3.13)

and we may assume that $U$ is open.

Clearly the minimality of $U$ implies

$\mathcal{H}^{n-1}(\partial U\cap\overline{\Omega})=P(U;\overline{\Omega})\leq
P(E;\Omega)=\mathcal{H}^{n-1}(\partial E\cap\Omega).$

Thus in order to prove (3.8), by Federer’s theorem [19, Theorem 16.2] it
suffices the existence of $\Theta^{\prime}>0$ so that

$\mathcal{H}^{n-1}(\overline{U}\cap\partial\Omega)\leq\Theta^{\prime}\mathcal{H}^{n-1}(\partial
U\cap\overline{\Omega}),$

which follows once, for another constant $\Theta^{\prime\prime}>0$

$\mathcal{H}^{n-1}(U\cap\partial\Omega)\leq\Theta^{\prime\prime}\mathcal{H}^{n-1}(\partial
U\cap\Omega)$ (3.14)

gets proven.

Recall that $\Omega$ is a John domain with center $x_{0}\in\Omega$. Then for
any $x\in U\cap\partial\Omega$, we claim the following:

The John curve ${\gamma}_{x}$ given by Lemma 3.3 crosses $\partial
U\cap\Omega$ at some point $y_{x}\in\partial U\cap\Omega$. (3.15)

Indeed, as $U$ is open, $x_{0}\notin B_{r}(x)$ and $x\in U\cap\partial\Omega$,
then ${\gamma}_{x}$ necessarily intersects $\partial U$. Besides, Lemma 3.3
tells that ${\gamma}_{x}\setminus\\{x\\}\subset\Omega$, which then implies
${\gamma}_{x}$ crossing $\partial U\cap\Omega$ at some point $y_{x}\in\partial
U\cap\Omega$. This gives (3.15).

As a consequence of (3.15), the definition of John curve (1.7) together with
(3.10), we conclude that, for any $x\in U\cap\partial\Omega$, there exists
$C_{0}=C_{0}(n,\,J)>0$ and $Q_{x}\in\mathscr{F}$ so that

$x\in C_{0}Q_{x}$ (3.16)

whenever we choose $Q_{x}$ containing $y_{x}$.

Step 2: Proof of (3.14). Now we are ready to show (3.14). Let

$\mathscr{S}:=\\{Q\in\mathscr{F}:Q=Q_{x}\text{ given by \eqref{Q cover
boundary} for some $x\in U\cap\partial\Omega$}\\}.$

Then it follows that

$U\cap\partial\Omega\subset\bigcup_{Q\in\mathscr{S}}2C_{0}Q$ (3.17)

Now since $\\{\frac{11}{10}Q\\}_{Q\in\mathscr{F}}$ has at most $C(n)$-overlaps
by (3.9), we conclude from (3.13) that

$\displaystyle\sum_{Q\in\mathscr{S}}\mathcal{H}^{n-1}\left(\partial
U\cap\frac{11}{10}Q\right)$
$\displaystyle\leq\sum_{Q\in\mathscr{F}}\mathcal{H}^{n-1}\left(\partial
U\cap\frac{11}{10}Q\right)\leq
C(n)\sum_{Q\in\mathscr{F}}\mathcal{H}^{n-1}(\partial U\cap Q)$
$\displaystyle=C(n)\mathcal{H}^{n-1}(\partial
U\cap\Omega)=C(n)\mathcal{H}^{n-1}(\partial_{*}U\cap\Omega).$ (3.18)

On the other hand, (3.12) gives

$\sum_{Q\in\mathscr{S}}\mathcal{H}^{n-1}\left(\partial
U\cap\frac{11}{10}Q\right)\geq c(n)\sum_{Q\in\mathscr{S}}l(Q)^{n-1}.$ (3.19)

In addition, (3.17) together with (1.9) yields

$\displaystyle\sum_{Q\in\mathscr{S}}l(Q)^{n-1}$ $\displaystyle\geq
c(n,\,J)\sum_{Q\in\mathscr{S}}\mathcal{H}^{n-1}\big{(}\partial\Omega\cap
2C_{0}Q\big{)}$ $\displaystyle\geq
c(n,\,J)\mathcal{H}^{n-1}(U\cap\partial\Omega)=c(n,\,J)\mathcal{H}^{n-1}(U\cap\partial_{*}\Omega).$
(3.20)

where the last equality follows again from Federer’s Theorem [19, Theorem
16.2], as $\Omega$ is of finite perimeter.

Combining (3.18), (3.19) and (3.20), we conclude (3.14), and hence (3.8) for
$E$. Therefore, $\Omega$ is admissible, and [27, Theorem 5.14.4] gives the
rest of the lemma. The proof is completed. ∎

Now we are ready to prove Theorem 1.4.

###### Proof of Theorem1.4.

Let $x_{0}\in\Omega$ be the John center of $\Omega$ and $\mathscr{F}$ be the
set of all Whitney cubes of $\Omega$. We choose $Q_{0}$ to be a cube in
$\mathscr{F}$ with the John center $x_{0}\in Q_{0}$. For any
$Q\in\mathscr{F}$, we denote by

$\hat{Q}:=\frac{11}{10}Q,\qquad u_{\hat{Q}}:={\hbox to0.0pt{\bf\kern
4.49997pt--\hss}\int}\ignorespaces_{\hat{Q}}u(x)dx.$

Step 1: $K=B$. We first prove (1.11) in case when $K$ is the unit ball $B$
under the Euclidean norm.

Step 1.1 : Estimate $|u_{\hat{Q_{i}}}-u_{\hat{Q_{j}}}|$ for any pair of cubes
$Q_{i},Q_{j}\in\mathscr{F}$ with $Q_{i}\cap Q_{j}\neq\emptyset$. Note that,
there exists a cube $R$ with

$R\subset\hat{Q}_{i}\cap\hat{Q}_{j}\quad\text{and}\quad\ell(R)=\frac{1}{20}\min\\{\ell(Q_{i}),\ell(Q_{j})\\}.$
(3.21)

Then by triangle inequality

$|u_{\hat{Q}_{i}}-u_{\hat{Q}_{j}}|\leq|u_{\hat{Q}_{i}}-u_{R}|+|u_{R}-u_{\hat{Q}_{j}}|.$

As (3.10) together with (3.21) gives $\ell(R)\geq\frac{1}{80}\ell(Q_{i})$, we
apply the triangle inequality and the $1$-Poincaré on $\hat{Q}_{i}$ to
conclude

$\displaystyle|u_{\hat{Q}_{i}}-u_{R}|$ $\displaystyle\leq{\hbox
to0.0pt{\bf\kern 4.49997pt--\hss}\int}\ignorespaces_{\hat{Q}_{i}}\left({\hbox
to0.0pt{\bf\kern
4.49997pt--\hss}\int}\ignorespaces_{R}|u(y)-u(z)|dz\right)dy\leq C(n){\hbox
to0.0pt{\bf\kern 4.49997pt--\hss}\int}\ignorespaces_{\hat{Q}_{i}}\left({\hbox
to0.0pt{\bf\kern
4.49997pt--\hss}\int}\ignorespaces_{\hat{Q}_{i}}|u(y)-u(z)|dz\right)dy$
$\displaystyle\leq C(n){\hbox to0.0pt{\bf\kern
4.49997pt--\hss}\int}\ignorespaces_{\hat{Q}_{i}}|u(y)-u_{\hat{Q}_{i}}|dy\leq
C(n)\ell(Q_{i}){\hbox to0.0pt{\bf\kern
4.49997pt--\hss}\int}\ignorespaces_{\hat{Q}_{i}}|Du(y)|dy.$ (3.22)

Likewise, we obtain a similar upper bound for $|u_{R}-u_{\hat{Q}_{j}}|$, thus

$|u_{\hat{Q}_{i}}-u_{\hat{Q}_{j}}|\leq C(n)\left(\ell(Q_{i}){\hbox
to0.0pt{\bf\kern
4.49997pt--\hss}\int}\ignorespaces_{\hat{Q}_{i}}|Du(y)|dy+\ell(Q_{j}){\hbox
to0.0pt{\bf\kern
4.49997pt--\hss}\int}\ignorespaces_{\hat{Q}_{j}}|Du(y)|dy\right).$ (3.23)

Step 1.2: Estimate $|Tu(x)-u_{\hat{Q}_{0}}|$ for any $x\in\partial\Omega$. Fix
$x\in\partial\Omega$. According to Lemma 3.3, there exists a $J$-John curve
$\gamma_{x}$ joining $x$ to $x_{0}$. Moreover, the definition of the John
curve (1.7) yields that, every Whitney cube
$Q\in\mathscr{A}_{x}:=\\{Q\in\mathscr{F}:Q\cap\gamma_{x}\neq\emptyset\\}$
satisfies

$\hat{Q}\subset B_{C_{1}\ell(Q)}(x)\cap\Omega\quad\text{ and }\quad x\in
C_{2}Q,$ (3.24)

for some constants $C_{1}=C_{1}(n,\,J)$ and $C_{2}=C_{2}(n,\,J)$.

Relabel $\mathscr{A}_{x}=\\{Q_{k}\\}_{k\in\mathbb{N}}$ so that $Q_{k}\cap
Q_{k+1}\neq\emptyset,k\in\mathbb{N}$ together with $Q_{k}\to x$ as
$k\to\infty$, and recall that $x_{0}\in Q_{0}$ Then (1.10) implies

$\lim_{k\to\infty}|u_{\hat{Q}_{k}}-Tu(x)|=0.$ (3.25)

Consequently, via (3.23), we get that

$|Tu(x)-u_{\hat{Q}_{0}}|\leq\sum_{k=0}^{+\infty}|u_{\hat{Q}_{k+1}}-u_{\hat{Q}_{k}}|\leq\sum_{k=0}^{+\infty}C(n)\ell(Q_{k}){\hbox
to0.0pt{\bf\kern 4.49997pt--\hss}\int}\ignorespaces_{\hat{Q}_{k}}|Du(y)|dy.$
(3.26)

Step 1.3: Final estiamte. Now we integrating both sides of (3.26) on
$\partial\Omega$ with respect to $\mathcal{H}^{n-1}$-measure and obtain that

$\int_{\partial\Omega}|Tu(x)-u_{\hat{Q}_{0}}|\,d\mathcal{H}^{n-1}(x)\leq
C(n)\int_{\partial\Omega}\sum_{Q_{k}\in\mathscr{A}_{x}}\ell(Q_{k})^{1-n}\int_{\hat{Q}_{k}}|Du(y)|\,dy\,d\mathcal{H}^{n-1}(x).$
(3.27)

For each $Q\in\mathscr{F}$, by (3.24) it refers to points $x$ contained in
$C_{1}C_{2}Q\cap\partial\Omega$. Then since

$\mathcal{H}^{n-1}(C_{1}C_{2}Q\cap\partial\Omega)\leq C(n,\,J)\ell(Q)^{n-1}$

accoridng to (1.9), by interchanging the integral and summation on the right-
hand side of (3.27) via Fubini’s theorem, we arrive at

$\displaystyle\int_{\partial\Omega}|Tu(x)-u_{\hat{Q}_{0}}|\,d\mathcal{H}^{n-1}(x)\leq$
$\displaystyle
C(n,\,J)\sum_{Q\in\mathscr{F}}\ell(Q_{k})^{1-n}\mathcal{H}^{n-1}(C_{1}C_{2}Q\cap\partial\Omega)\int_{\hat{Q}_{k}}|Du(y)|\,dy$
$\displaystyle\leq$ $\displaystyle
C(n,\,J)\sum_{Q\in\mathscr{F}}\int_{\hat{Q}_{k}}|Du(y)|\,dy\leq
C(n,\,J)\int_{\Omega}|Du(y)|\,dy.$

Thus we conclude (1.11) holds when $K=B$.

Step 2: General case. Since $\Omega$ is a $J$-John domain under
$\|\cdot\|_{K}$, it is also $\hat{J}$-John under the standard Euclidean norm
$|\cdot|$ with $\hat{J}=J\frac{M_{K}}{m_{K}}$. Therefore, by gathering (1.4),
the assumption (3.1) and the conclusion from Step 1, we have

$\displaystyle\int_{\partial\Omega}|Tu(x)-u_{\hat{Q_{0}}}|\|\nu_{\Omega}\|_{*}d\mathcal{H}^{n-1}(x)\leq
M_{K}\int_{\partial\Omega}|Tu(x)-u_{\hat{Q_{0}}}|d\mathcal{H}^{n-1}(x)$
$\displaystyle\leq C(n,\,\hat{J})M_{K}\int_{\Omega}|Du(w)|dw\leq
C(n,\,\hat{J})\frac{M_{K}}{m_{K}}\int_{\Omega}\|Du(w)\|_{*}dw$
$\displaystyle\leq
C\left(n,\,J,\,\frac{M_{K}}{m_{K}}\right)\int_{\Omega}\|Du(w)\|_{*}dw\leq
C\left(n,\,J\right)\int_{\Omega}\|Du(w)\|_{*}dw.$

The proof is completed. ∎

##  Appendix A Proof of the Selection Principle.

###### Proof of Proposition 1.5.

Step 1: We first claim that for any $x\in\mathbb{R}^{n}$,
$B_{\|\cdot\|_{K},1}(x)$ is the unique minimizer up to translations of the
problem:

$\min\\{P_{K}(U)+\Lambda\big{|}|U|-|K|\big{|}:U\subset\mathbb{R}^{n}\\}\quad\text{
for $\Lambda>n$}.$ (A.1)

Indeed, by isoperimetric inequality, we may replace $U$ in (A.1) with the ball
$B_{\|\cdot\|_{K},r}(x)$. Then (A.1) is equivalent to find the point where the
minimum of $h(r):=nr^{n-1}+\Lambda|r^{n}-1|$ can be reached. Since $h$ has a
unique minimum when $r=1$ in case $\Lambda>n$, our claim holds.

Step 2: Replace $E_{k}$ by a $(\Lambda+1,R_{0})$-minimizer. Let $\Lambda=n+1$
and $R_{0}=10$. To set up the new set, for every $k\in\mathbb{N}^{+}$ we
consider a minimizer $F_{k}^{\prime}$ of the following problem:

$\min\Big{\\{}P_{K}(U)+\big{|}|U\Delta K|-|E_{k}\Delta
K|\big{|}+\Lambda\big{|}|U|-|K|\big{|}:U\subset
B_{\|\cdot\|_{K},R_{0}}\Big{\\}},$ (A.2)

where $\Lambda>n$ is fixed. Then by [19, Theorem 12.26], up to passing to a
subsequence, we may assume that there is a set $F$ so that

$|F_{k}^{\prime}\Delta F|\to 0\quad\text{ as }k\to+\infty,\qquad
P_{K}(F)\leq\liminf_{k\to+\infty}P_{K}(F_{k}^{\prime}).$ (A.3)

As a result, recalling $E_{k}\to K$ as $k\to+\infty$, we use (A.2) and (A.3)
to obtain the minimality of $F$, that is, for any $U\subset
B_{\|\cdot\|_{K},R_{0}}$ with finite perimeter,

$\displaystyle P_{K}(F)+|F\Delta
K|+\Lambda\big{|}|F|-|K|\big{|}\leq\liminf_{k\to+\infty}P_{K}(F_{k}^{\prime})+|F_{k}^{\prime}\Delta
K|+\Lambda\big{|}|F_{k}^{\prime}|-|K|\big{|}$
$\displaystyle\leq\liminf_{k\to+\infty}P_{K}(F_{k}^{\prime})+(\big{|}|F_{k}^{\prime}\Delta
K|-|E_{k}\Delta K|\big{|}+|E_{k}\Delta
K|)+\Lambda\big{|}|F_{k}^{\prime}|-|K|\big{|}$
$\displaystyle\leq\liminf_{k\to+\infty}P_{K}(U)+\big{|}|U\Delta
K|-|E_{k}\Delta K|\big{|}+|E_{k}\Delta K|+\Lambda\big{|}|U|-|K|\big{|}$
$\displaystyle\leq P_{K}(U)+|U\Delta K|+\Lambda\big{|}|U|-|K|\big{|}.$ (A.4)

Further recalling (A.1) and that $K$ is the unique solution of

$\min\\{|U\Delta K|:U\in B_{\|\cdot\|_{K},R_{0}}\\},$

it implies that $K$ is the unique solution of

$\min\\{P_{K}(U)+|U\Delta K|+\Lambda\big{|}|U|-|K|\big{|}:U\subset
B_{\|\cdot\|_{K},R_{0}}\\}.$

As a result, from (A) it follows that $F=K$.

Next we show that $F_{k}^{\prime}$ are all $(\Lambda+1,R_{0})$-minimizers and
that their boundary converge to $\partial K$ in Hausdorff metric. To show
this, from Lemma 2.3 and $F_{k}^{\prime}\subset B_{\|\cdot\|_{K},R_{0}}$, it
suffices to prove the former.

To this aim, we optionally choose a ball $B_{\|\cdot\|_{K},r}(x)$ with $x\in
F_{k}^{\prime}$ and $r\in(0,R_{0})$, and a set $U$ satisfying
$F_{k}^{\prime}\Delta U\subset\subset B_{\|\cdot\|_{K},r}(x)$. Then we have
two cases:

Case 1: $U\subset B_{\|\cdot\|_{K},R_{0}}$. By using (A.2) and the triangle
inequality in sequence, we have

$\displaystyle P_{K}(F_{k}^{\prime})$ $\displaystyle\leq
P_{K}(U)+\big{|}|U\Delta K|-|E_{k}\Delta
K|\big{|}-\big{|}|F_{k}^{\prime}\Delta K|-|E_{k}\Delta
K|\big{|}+\Lambda\big{|}|U|-|K|\big{|}-\Lambda\big{|}|F_{k}^{\prime}|-|K|\big{|}$
$\displaystyle\leq P_{K}(U)+\big{|}|U\Delta K|-|F_{k}^{\prime}\Delta
K|\big{|}+\Lambda\big{|}|U|-|F_{k}^{\prime}|\big{|}$ $\displaystyle\leq
P_{K}(U)+(1+\Lambda)|U\Delta F_{k}^{\prime}|.$ (A.5)

Case 2:$|U\setminus B_{\|\cdot\|_{K},R_{0}}|>0$. Let $U^{\prime}:=U\cap
B_{\|\cdot\|_{K},R_{0}}$. Since $F_{k}^{\prime}\cup U^{\prime}\subset
B_{\|\cdot\|_{K},R_{0}}$ holds from the definition of $U^{\prime}$ and (A.2),
it leads to $|F_{k}^{\prime}\setminus U^{\prime}|=|F_{k}^{\prime}\setminus U|$
so that

$|U^{\prime}\Delta F_{k}^{\prime}|=|U^{\prime}\setminus
F_{k}^{\prime}|+|F_{k}^{\prime}\setminus U^{\prime}|\leq|U\setminus
F_{k}^{\prime}|+|F_{k}^{\prime}\setminus U|=|U\Delta F_{k}^{\prime}|.$

Consequently, again using $U^{\prime}\subset B_{\|\cdot\|_{K},R_{0}}$, we
obtain from the consequence of case 1 that

$P_{K}(F_{k}^{\prime})-P_{K}(U^{\prime})\leq(\Lambda+1)|U^{\prime}\Delta
F_{k}^{\prime}|\leq(\Lambda+1)|U\Delta F_{k}^{\prime}|.$ (A.6)

Moreover, the isoperimetric inequality applied to $U\cup
B_{\|\cdot\|_{K},R_{0}}$ tells that there exists a ball $B_{\|\cdot\|_{K},R}$
with $R>R_{0}$ and $|U\cup B_{\|\cdot\|_{K},R_{0}}|=|B_{\|\cdot\|_{K},R}|$ so
that

$P_{K}(B_{\|\cdot\|_{K},R_{0}})<P_{K}(B_{\|\cdot\|_{K},R})\leq P_{K}(U\cup
B_{\|\cdot\|_{K},R_{0}}).$ (A.7)

and then (A.7) gives

$P_{K}(U^{\prime})-P_{K}(U)=P_{K}(B_{\|\cdot\|_{K},R_{0}})-P_{K}(U\cup
B_{\|\cdot\|_{K},R_{0}})\leq 0,$

which, combined with (A.6), yields that

$\displaystyle P_{K}(F_{k}^{\prime})-P_{K}(U)$
$\displaystyle=\big{(}P_{K}(F_{k}^{\prime})-P_{K}(U^{\prime})\big{)}+\big{(}P_{K}(U^{\prime})-P_{K}(U)\big{)}\leq(\Lambda+1)|U\Delta
F_{k}^{\prime}|.$ (A.8)

Step 3: $F_{k}^{\prime}$ are John whenever $k$ is sufficiently large.  We
claim that up to passing to a subsequence, for the same $\delta$ in Lemma 3.1,
whenever $k$ is sufficiently large,

$\partial F_{k}^{\prime}\subset(1+\delta)K\setminus(1-\delta)K$ (A.9)

holds such that

$(1-\delta)K\subset F_{k}^{\prime}\subset(1+\delta)K$ (A.10)

Indeed, the uniform boundness of $F_{k}^{\prime}$ and (A.9) ensure that
$F_{k}^{\prime}\subset(1+\delta)K$. We further suppose that there is a point
$z\in(1-\delta)K$ with $z\notin F_{k}^{\prime}$. Then any curve
$\gamma\subset(1-\delta)K$ joining $z$ to $\partial(1-\delta)K$ satisfies
$\gamma\cap\overline{F_{k}^{\prime}}=\emptyset$ from (A.9) and the
connectivity of $\gamma$. As a result,
$(1-\delta)K\subset\mathbb{R}^{n}\setminus F_{k}^{\prime}$ so that

$0\leftarrow|F_{k}^{\prime}\Delta K|\geq|K\setminus
F_{k}^{\prime}|\geq|(1-\delta)K|$

which yields the contradiction. Hence, $(1-\delta)K\subset F_{k}^{\prime}$ so
that (A.10) holds.

As a result, combining (A.10) and Lemma 3.1, $F_{k}^{\prime}$ are
$(\Lambda+1,R_{0})$-minimizers so that $F_{k}^{\prime}$ are all $J$-John
domains with $J=J(n)$.

Step 4: Properly scale $F_{k}^{\prime}$. Recalling (1.12), we have

$P_{K}(E_{k})-P_{K}(K)\leq\beta_{k}|E_{k}\Delta K|^{2},$

where $\beta_{k}\in(0,2^{-k}]$, since we have taken subsequence for $E_{k}$
and $F_{k}^{\prime}$ multiple times. The minimality (A.2) of $F_{k}^{\prime}$
gives

$\displaystyle
P_{K}(F_{k}^{\prime})+\Lambda\big{|}|F_{k}^{\prime}|-|K|\big{|}+\big{|}|F_{k}^{\prime}\Delta
K|-|E_{k}\Delta K|\big{|}\leq P_{K}(E_{k})\leq P_{K}(K)+\beta_{k}|E_{k}\Delta
K|^{2}$ (A.11)

and then the minimality (A.1) of $K$ further provides

$P_{K}(K)+\beta_{k}|E_{k}\Delta K|^{2}\leq
P_{K}(F_{k}^{\prime})+\Lambda\big{|}|F_{k}^{\prime}|-|K|\big{|}+\beta_{k}|E_{k}\Delta
K|^{2}.$ (A.12)

Hence, by comparing the left hand side of (A.11) and the right hand side of
(A.12), we obtain that $\big{|}|F_{k}^{\prime}\Delta K|-|E_{k}\Delta
K|\big{|}\leq\beta_{k}|E_{k}\Delta K|^{2}$. This, by gathering the assumption
$E_{k}\Delta K\to 0$ and $\beta_{k}\to 0$, leads to

$\lim_{k\to+\infty}\frac{|F_{k}^{\prime}\Delta K|}{|E_{k}\Delta K|}=1.$ (A.13)

To achieve the proof we need to scale $F_{k}^{\prime}$ appropriately. Assume
that $F_{k}=\lambda_{k}F_{k}^{\prime}$, where $\lambda_{k}$ satisfies
$|F_{k}|=|K|$. Observe that $\lambda_{k}\to 1$ holds by $F_{k}^{\prime}\to K$.
Then via $\lim_{k\to+\infty}P_{K}(F_{k}^{\prime})=P_{K}(K)$ given by (A.11),
we have $\lim_{k\to+\infty}P_{K}(F_{k})=P_{K}(K)$. Hence,
$P_{K}(F_{k})/|F_{k}|<\Lambda$ holds whenever $k$ is sufficiently large, since
$\lim_{k\to\infty}P_{K}(F_{k})/|F_{k}|=P_{K}(K)/|K|=n<\Lambda$. Consequently,
for $k$ large enough we have

$\displaystyle|P_{K}(F_{k})-P_{K}(F_{k}^{\prime})|$
$\displaystyle=P_{K}(F_{k})|1-\lambda_{k}^{1-n}|\leq
P_{K}(F_{k})|1-\lambda_{k}^{-n}|$
$\displaystyle<\Lambda|F_{k}||1-\lambda_{k}^{-n}|=\Lambda\big{|}|F_{k}|-|F_{k}^{\prime}|\big{|}.$
(A.14)

which, together with (A.11), yields that

$P_{K}(F_{k})\leq
P_{K}(F_{k}^{\prime})+\Lambda\big{|}|F_{k}|-|F_{k}^{\prime}|\big{|}=P_{K}(F_{k}^{\prime})+\Lambda\big{|}|K|-|F_{k}^{\prime}|\big{|}\leq
P_{K}(K)+\beta_{k}\big{|}E_{k}\Delta K\big{|}^{2}$ (A.15)

Now we recall that (A.13) implies that $|E_{k}\Delta K|^{2}<2|F_{k}\Delta
K|^{2}$ whenever $k$ is large enough, from which (A.15) gives (1.13) by
setting $\alpha_{k}:=2\beta_{k}$. To conclude, $F_{k}$ are the desired
$J$-John domains. ∎

## References

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