# Questions tagged [geometric-measure-theory]

Questions about geometric properties of sets using measure theoretic techniques; rectifiability of sets and measures, currents, Plateau problem, isoperimetric inequality and related topics.

599
questions

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### Measure estimates of $\delta$-neighbourhood of compact sets

I am interested in the estimating from above the measure of a compact set $K$ by a sequence of sets $K_n$, converging to it in the Hausdorff metric. As such I am looking for known conditions that give ...

1
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1
answer

64
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### When is the mode of a stochastic process a better statistic than the mean?

This is a soft question.
I've been interested in Onsager-Machlup theory recently. Essentially, the Onsager-Machlup function serves the role of a density but it can exist on non locally compact spaces.
...

4
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92
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### Density of smooth function in the calculus of variations

In the non-convex calculus of variations, in the context of non-linear elasticity, the following classes of mappings $u:\Omega\to\mathbb{R}^n$, $\Omega\subset\mathbb{R}^n$, were introduced by John ...

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### Are Carnot groups ever CAT(𝜅) spaces?

Let $G$ be a free Carnot group of homogeneous dimension $d$, equipped with the Carnot–Carathéodory metric. Is $(G,d)$ ever $\operatorname{CAT}(\kappa)$ for some $\kappa\in \mathbb{R}$?

11
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247
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### Does every smooth map of rank at most d factor through a d-manifold?

Suppose $d≥0$, $m≥0$, $n≥0$, and $\def\R{{\bf R}} f\colon \R^m→\R^n$ is a smooth map
whose rank at any point of $\R^m$ is at most $d$.
Here and below, smooth means infinitely differentiable.
Can we ...

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0
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68
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### Distortion estimates to control Hausdorff measure of a curve

I am studying the paper Blumenthal - Statistical properties for compositions of standard maps with increasing coefficent.
I have a problem to understand how the distortion estimates are used. The ...

5
votes

1
answer

132
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### Is every set with finite $\mathcal{H}^{n-1}$ measure a set of locally finite perimeter?

Given a measurable set $E \subset \mathbb{R}^d$, with $\mathcal{H}^{d-1} (\partial E) < +\infty$, is it true in general that $E$ is a set of locally finite perimeter? that is, is it true that $\...

8
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1
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454
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### On functions with strict Lipschitz constant

We say a measurable subset $S$ of $\mathbb R^n$ is measure dense if for every open set $U \subset \mathbb R^n$, $U \cap S$ is of positive Lebesgue measure.
Let $n \geq 2$, and let $f: \mathbb R^n \to \...

6
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1
answer

229
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### Lipschitz property of the symmetric rearrangement

I'm currently reading Talenti's paper "Best constant in Sobolev inequality" and am rather stuck on an argument on pg 363 (or pg 11 if you're reading the pdf). In this section of the paper, ...

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1
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62
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### Approximating the probability of a half-space using random Voronoi diagrams

Fix a half-space $H = \{x_1 \geq 0: ~ (x_1,\dots,x_n) \in \mathbb{R}^n\}$. Let $p$ be a distribution with support in $\mathbb{R}^n$. I am interested in the following way of estimating the weight $p(H) ...

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2
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117
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### Distribution of the support function of convex bodies: beyond mean width

Let $K$ be a symmetric convex body in $\mathbb{R}^n$ (that is the unit ball of a norm). Let $h_K$ be its support function, that is $h_K(u) = \sup_{x \in K}\langle x,u \rangle$. The quantity $w(K) = \...

2
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111
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### Approximating a uniformly elliptic function of bounded variation by Lipschitz functions

Let $\sigma: \mathbb R^d \to \mathbb R^{d \times d}$ be of bounded variation.
Suppose $\sigma$ is uniformly elliptic, in the sense that there exists some constant $C > 0$ such that $\xi^{T} \sigma(...

2
votes

1
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### Examples of "almost" Ahlfors regular measures

Let $\mu$ be a Borel probability measure on $\mathbb{R}^n$ such that there are $c,C,d,D>0$ satisfying: for every $x \in \mathbb{R}^n$ and every $r>0$
$$
c r^d \leq \mu(B(x,r)) \leq Cr^D.
$$
Let'...

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0
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68
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### Volume of intersection of $\varepsilon$-neighbourhoods of graphs

Let $n \geq 3$, and suppose $f, g: \mathbb R^{n-1} \to \mathbb R$ are smooth functions satisfying the following transversality condition:
Denote by $\Gamma(f)$ and $\Gamma(g)$ the graphs of $f$ and $g$...

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37
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### Minimal condition on set for an optimisation problem

We fix $\Omega \subset \mathbb{R}^{2}$ an open set. My question is: what are the minimum conditions we need on $E \subset \Omega$ such that the following optimisation problem:
$$
\sup\{ \int_{E}(\...

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0
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69
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### Lower bound estimate for the sum $\sum \text{diam}(U)^d$ over all countable covers of a cube

This question is inspired from the definition of Hausdorff measure. Let $C$ be a closed unit hypercube in $\mathbb R^d$ (side length equal to one, including boundary. The cube itself is at top ...

6
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123
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### Topological properties of the dual of differential forms

Notation:
$U \subset R^n$, bounded open set
$D^k(U) = \{ \omega : U \to \Lambda^k R^n : \text{compactly supported and infinitely differentiable \}}$
$D_k(U) = D^k(U)'$ is the topological dual space (...

0
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1
answer

116
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### When does $C_b(X)$ admit a Schauder Basis?

Let $(X,d)$ be a separable and connected metric space. My question is rather short and to the point: do there exist $\{x_n\}_{n=0}^{\infty}\subseteq X$ such that
$$
\left\{d(x_n,\cdot)-d(x_0,\cdot)\...

2
votes

1
answer

127
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### Disjoint union of closed sets

It is well-known that $[0,1]$ is not a nontrivial disjoint union of closed intervals -- e.g.:
https://math.stackexchange.com/questions/1195179/the-interval-0-1-is-not-the-disjoint-countable-union-of-...

9
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0
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409
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### Measure theoretic boundary in arbitrary codimension

NOTE: I had initially posted here the same question you can find on math.stackexchange at Boundary in the sense of currents VS measure theoretic boundary in arbitrary codimension. Later, I decided to ...

18
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4
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### Generalized Stokes' theorem

In the Wikipedia article on Stokes' theorem the following claim is advanced without any references given:
The main challenge in a precise statement of Stokes' theorem is in defining the notion of a ...

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1
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93
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### A question about pushforward measures and Peano spaces

Specifically my question is the following: Let $P$ be a Peano space. If $(P,\sigma,\mu)$ and $(P,\sigma,\nu)$ are both nonatomic probability measures, does there exist a continuous function $f:P\to P$ ...

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151
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### A locally compact, complete metric space in which the closure of open balls coincide with the closed ball is Heine-Borel

I saw the following result stated without a proof in a paper about the isometry group of metric measure spaces:
Let $X$ be a locally compact, complete metric space such that for all $x \in X$ and $R &...

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165
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### approximation of currents

Let $M$ be a closed Riemannian manifold of dimension $d$. Let $d \alpha$ be a smooth exact $p$-form. We define a current $T_{d \alpha}$ as follows : for any smooth $(d-p)$-form $\beta$ we set
$$ T_{d \...

2
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0
answers

65
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### Estimate of Wasserstein distance and flow of vector fields under particular assumptions

Let $\mu$ be a compactly supported absolutely continuous probability measure. Let $v,u$ be Lipschitz vector fields. For a vector field $w$ recall that $\Phi_t^w$ denotes its flow.
A classical estimate ...

2
votes

1
answer

146
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### A question about pushforward measures and continuous Borel isomorphisms

It is fairly well known that if $\mu$ and $\nu$ are nonatomic measures on the standard Borel spaces $(X,B)$ and $(Y,C)$ such that $\mu(X)=\nu(Y)$. If $X$ and $Y$ are uncountable, then there exists a ...

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1
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125
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### Does a submanifold of nonzero codimension have measure zero under the product of non atomic measures?

Let $A$ be a non atomic measure on $\mathbb R$. Consider the product measure $\mu := A \times \dots \times A$ on $\mathbb R^n$.
Question: Let $M$ be a $n-1$ dimensional smooth submanifold of $\mathbb ...

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0
answers

26
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### Does a constant-volume continuous deformation imply the existence of a volume-preserving continuous deformation?

For measurable ${\bf A} \subset R^{n}$, let $\mu({\bf A})$ be the $n$-dimensional measure of $\bf A$.
Let ${\bf B} \subset R^{n}$ be homeomorphic to the closed unit ball. Let $\Gamma:{\bf B} \times [0,...

3
votes

0
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### Hausdorff measure of the unit ball of a norm on $\mathbb{R}^n$ is a universal constant

In [1], Kirchheim proved the area formula for Lipschitz maps $f\colon \mathbb{R}^n\to X$ where $X$ is an arbitrary metric space, using the notion of metric differentiability. The metric derivative of $...

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votes

1
answer

144
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### Example where concentration of measure fails nontrivially

A metric probability space $(X, \mu, \rho)$, i.e., a complete separable metric space with a probability measure on its Borel sets, is said to satisfy (Gaussian) concentration of measure property if ...

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1
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145
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### When are Wasserstein spaces $CAT(\kappa)$?

Let $(X,d)$ be a complete and separable metric space and, for $1\leq p<\infty$, let $(\mathcal{P}_p(X,d),W_p)$ be the $p$-Wasserstein space on $(X,d)$. For which $p$ and $(X,d)$ is $(\mathcal{P}_p(...

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0
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70
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### Absolute continuity of joint distribution if all marginals in any basis are absolutely continuous

Consider a probability distribution $\nu$ on $(x,y)\in\mathbb{R}^2$. I know that the absolute continuity of the marginals on $x$ and $y$ is not sufficient to imply the absolute continuity of $\nu$, ...

4
votes

1
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153
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### On the set on which $|Df|$ is maximal for Lipschitz $f$

Let $f: \mathbb R^n \to \mathbb R$ be a Lipschitz continuous function with strict Lipschitz constant $L > 0$.
That is, $|f(x) - f(y)| < L|x - y|$ for all $x \neq y$ in $\mathbb R^d$.
Question: ...

8
votes

1
answer

305
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### Calderon-Zygmund decomposition on manifolds?

The classical Calderon-Zygmund decomposition says that if $f\geq 0$ is $L^1$ on a cubes $B$, with average value $\alpha$, then there is a sequence of disjoint cubes $B_j$, such that the average of $f$ ...

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126
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### Laplacian spectrum and measured Gromov-Hausdorff convergence of Riemannian manifolds with boundary

In the paper "Collapsing of Riemannian manifolds and eigenvalues of Laplace operator" by Kenji Fukaya, it is proven that the spectrum of the Laplacian is continuous with respect to measured ...

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39
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### Measure and other properties of nodal domains of Laplacian

Let $(\phi_k,\lambda_k)$ be the couple of eigenfunctions and eigenvalues of the the Laplacian operator on $\Omega \subset \mathbb R^n$.
The nodal set of $\phi_k$ is the set $$\mathcal N_k = \{x \in \...

3
votes

1
answer

109
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### Is there a classification of the first geodesic nets?

A geodesic net is an embedding of a multigraph $(V,E)$ into a Riemannian manifold $(M,g)$, so that the vertices are mapped to points of $M$ and the edges to geodesics connecting them. Additionally, ...

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votes

2
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206
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### Wasserstein convergence of "series expansion'' of probability measure

Let $X$ be a Polish space and let $(\mu_i)_{i=1}^{\infty}$ be a sequence of probability measures in the Wasserstein space $\mathcal{P}(X)$ on $X$. Let $(\beta_i)_{i=1}^{\infty}$ be a summable ...

9
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342
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### Tiling the plane with finitely many congruent pieces

Suppose $A_1,\dots,A_n$ are measurable subsets of the plane that are all related by rigid motions such that $|(A_1 \cup \dots \cup A_n)^c| = 0$ and $|A_i \cap A_j| = 0$ for all $1 \leq i < j \leq n$...

2
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80
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### Can we define surface integral on 'bad surface'?

We can define a surface integral on a piecewise smooth surface, but if the surface is not piecewise smooth can we use measure theory to generalize the definition of surface integral? And does Stokes ...

2
votes

1
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139
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### Local dimension of measures

For a Borel prob measure $\mu$ in $\mathbb{R}^n$, define the local dimension of $\mu$ at $x$ by
$$
{\rm dim}_*(\mu, x)=\liminf_{r\to 0} \frac{\log \mu(B(x,r))}{\log r}, {\rm dim}^*(\mu, x)=\limsup_{r\...

1
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0
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56
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### Is it possible to define the trace of a function over a rectifiable set?

Let $\Omega$ be a bounded open set with smooth boundary and $E$ a set of finite perimeter in $\Omega$, i.e. $$P(E;\Omega)=\left\{\int_E\text{div}\: T\:dx:T\in C^\infty_c(\Omega;\mathbb{R}^n), |T|\leq1\...

5
votes

1
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170
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### Set where the speed of convergence is uniform in Lebesgue's density theorem

Let $B \subset \mathbb R^n$ be the unit ball.
Consider a Borel measurable set $E \subset B$ with positive Lebesgue measure $|E|>0$ (say $|E| = |B|/2$).
Then, Lebesgue's density theorem, says that ...

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86
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### The behavior of an integral related to the inward normal vector near a point of the boundary of a domain

Inspired by this Q&A, I am asking for what kind of non-smooth domains $D$ the following limit
$$
\lim_{r \to 0}\frac{1}{m(D \cap B(x,r))}\int\limits_{D \cap B(x,r)}\frac{z-x}{r}\,m(dz)
$$
where
$...

0
votes

0
answers

52
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### Can iterative application of ham sandwich cuts form streamlines of an ODE?

It has been known that given two probability distributions $\mu_1$ and $\mu_2$ (let us say, they are smooth for simplicity), there is a hyperplane that divides the domain into two regions (denoted as $...

3
votes

0
answers

86
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### When is the least-area surface unique?

Let $M^{n-1}$ be a smooth closed manifold, embedded into the round sphere $\mathbf{S}^n$ via a regular map $\Phi$. Using tools from geometric measure theory, one can prove the existence of a $n$-...

5
votes

0
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150
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### How far can a continuous, almost everywhere differentiable function be from being a Sobolev function?

Let $\Omega$ be the open unit ball in $\mathbb R^n$. Consider the set $\mathcal D$ of continuous functions $f:\Omega \to \mathbb R$ that are differentiable a.e, and with $|\nabla f| \leq 1$ wherever $...

1
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0
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93
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### Hausdorff dimension of a compact Lie group [closed]

Let $G$ be a compact Lie group (for simplicity assume $G=SO(3)$). Equip $G$ with a left-invariant Riemannian metric and let $m$ be left-invariant Haar measure on $G$.
Now that $G$ is a metric space ...

0
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0
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271
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### Compact connected Riemannian manifolds are Ahlfors regular metric space

Let $(M,g)$ be a compact connected $n$-dimensional Riemannian manifold; let $(X,d)$ denote its associated metric (length) space. A comment on the original formulation of this post mentioned that $(X,...

2
votes

1
answer

224
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### Continuity of the perimeter of level sets w.r.t. level function

Working with the level set method introduced by Osher & Sethian in shape optimization I came across a simple question that I did not succeed to prove. It mainly asserts that the perimeter of the ...