Questions tagged [isoperimetric-problems]

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Sphere with bounded curvature

Let $V$ be a body in $\mathbb{R}^3$ bounded by a smooth sphere with principle curvatures at most 1 (by absolute value). Is it true that $$\mathop{\rm vol} V\ge \mathop{\rm vol} B,$$ where $B$ denotes ...
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Nonlocal perimeter of level sets

Let $u \in W^{s,1}(B)$ be given and $k < l$ be two numbers, then I am looking for a way to bound the following term from above. Here $B$ is the euclidean ball. $$ \int_{B: u < k} \int_{B:u>l} ...
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Integrating a function of distance between a set and its neighbourhood

I am aware of the isoperimetric inequality, which states that if you fix the Lebesgue measure of a measurable set $A \subset \mathbb R^d. d \geq 2$ then the smallest possible value of the perimeter of ...
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$\newcommand\v{\operatorname{vol}_d(C}$Compact subsets of $ℝ^d$ which maximize $\inf_{|v|\le1}\dfrac{\v\cap(𝜀v+C))}{\v)}$ for fixed $\v)$ and $𝜀>0$

Let $\operatorname{vol}_d$ be the volume measure on $\mathbb R^d$ and let $B_d$ be the unit-ball. For $\varepsilon \ge 0$ and a compact subset $C$ of $\mathbb R^d$ with $\operatorname{vol}_d(C)>0$, ...
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Nearest point is always regular for isoperimetric hypersurfaces

In his paper "Paul Levy's Isoperimetric Inequality" (published as appendix C in Metric Structures for Riemannian and Non-riemannian Spaces), Gromov claims that if $H$ is a minimal $n$-...
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Isoperimetric inequality for general metric space

Consider some space $\mathcal{S}$ with metric $d$ and measure $\mu$. For arbitrary set $H$ denote the $v$-bound of $H$ by $\delta_v(H):= \{x \mid x \notin H: \exists y \in H \text{ s.t. } d(x,y) \le v ...
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1 answer
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Isoperimetric inequality for domains in the exterior of a precompact open set in Riemannian manifold

Fix $n\geq 2$ and let $$\mathbb{H}^{n}=\mathbb{R}_{+}\times \mathbb{S}^{n-1}$$ be the hyperbolic space, so that any point $x\in \mathbb{H}^{n}$ can be represented in polar coordinates $x=(r, \theta)$, ...
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Isoperimetric inequality for exterior domains on $\mathbb{H}^{n}$

Fix $n\geq 2$ and let $\mathbb{H}^{n}=\mathbb{R}_{+}\times \mathbb{S}^{n-1}$ be the hyperbolic space be defined as a Riemannian manifold equipped with the Riemannian metric $$g=dr^{2}+\sinh^{2}rd\...
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Characterization of planar domains onto which a unit disk can be mapped with constant singular values

It can be shown that there are (smoothly bounded, Jordan) domains $E\subset \mathbb{R}^2$ which are $\textit{not}$ images of mappings $f$ from the unit disk (or any other planar domain), such that $\...
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Asymptotic optimal sphericity

How quickly does maximum sphericity of polyhedra with $n$ faces approach 1 as $n→∞$? I can show that sphericity $1 - \frac{5 \sqrt{3} π}{27n} - O(n^{-3/2})$ is possible. Is this, especially $O(n^{-3/...
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Isoperimetric type inequality in $\mathbb{R}^2$

Fix $L \in (0,\infty)$ and consider $\mathcal{C}_L$ defined as follows: \begin{align*} \mathcal{C}_L := \{ \gamma:[0,1] \rightarrow \mathbb{R}^2 |~ \gamma \text{ is smooth and length($\gamma$)$=L$ }\}....
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Does it holds that the $L^{\infty}$ norm of the support function of a convex body is minimal on balls with the same volume? [closed]

I was wondering if the following inequality holds. Let $K$ be a convex body of $\mathbb{R}^n$ and let us denote by $h_K$ its support function, defined as, for $x\in\mathbb{R}^n$ $$ h_K(x)={\max}\{x\...
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An Indepth Look at Isoperimetry in the Cayley Graph Generated by All Transpositions

Let $\Omega_n$ denote the symmetric/permutation group on $n$ objects. Let $T_n \subseteq \Omega_n$ denote the set of transpositions. Drop the $n$-subscripts. Define the Cayley graph $G = (\Omega, E)$ ...
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Lower-bound for $\underset{p \le \gamma_d(A) \le q}{\inf} \gamma(A^\epsilon)$, where $\gamma_d$ is the standard gaussian distribution on $\mathbb R^d$

Let $\gamma_d = \gamma_1 \otimes \ldots \otimes \gamma_1$ be the standard Gaussian distribution on $\mathbb R^d$, where $d$ is a large positive integer. Given $\epsilon \ge 0$ and a measurable $A \...
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Lower-bound on Sobolev norm of function on $(d-1)$-dimensional sphere, whose sign has been fixed at $n$ points

Let $\mathbb S_{d-1} := \{x \in \mathbb R^d \mid x^\top x = 1\}$ be $(d-1)$-dimensional sphere in $\mathbb R^d$ and let $\sigma_d$ be the uniform distribution on $\mathbb S_{d-1}$. Let $x_1,\ldots,x_n$...
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7 votes
3 answers
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When is perimeter continuous under Hausdorff convergence?

It is known that the perimeter is lower semicontinuous for the convergence of sets. Two variants are widely known: (Golab's theorem) in $\Bbb{R}^2$ if the sets $\Omega_n$ converge to $\Omega$ in the ...
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"Isoperimetric inequality" for self intersecting closed surfaces?

As the title suggests, I am trying to find something like an isoperimetric inequality for smooth immersions of the 2-sphere into $\mathbb{R}^3$ that relates the surface area to the enclosed 3d-volume. ...
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Perimeter decreases under intersection with half spaces

The main thing i need to prove is the following assertion: Let $E\subset R^N$ be a set of finite perimeter and $H=\{x\in R^N : x\cdot e < t \}$ for $t\in R$ and $e\in S^{N-1}$. Then prove that $$ ...
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1 vote
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Cheeger constant of truncated hypercube

Look at the $d$-dimensional hypercube and truncate it. This means one replaces each vertex by a cycle (of length $d$) in such a way the the new graph is 3-regular. Question 1: What is the asymptotic ...
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3 votes
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160 views

Isoperimetric inequality for analytic functions on an annulus

Let $f$ be an anylytic function on the unid disk $|z|<1$. It is well known that $$\left (\int_0^{2\pi}f(e^{i\theta})d \theta \right)^2 \geq 4\pi \iint_{|z|<1} |f(r e^{i\theta})|^2r dr d \theta.$...
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3 votes
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Minimizing expected mutual distances in spherical regions

Suppose I take the unit sphere in $d$ dimensions, and I take some subset $A$ of the sphere of fixed relative volume $V$. Now from this set $A$ I draw two vectors, uniformly at random, and I look at ...
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Is there a volume-preserving diffeomorphism of the disk with prescribed singular values?

This is a cross-post. While working on a variational problem, I have reached to the following question. Let $0<\sigma_1<\sigma_2$ satisfy $\sigma_1\sigma_2=1$, and let $D \subseteq \mathbb{R}^2$...
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1 vote
1 answer
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Area of a surface confined by a sphere II

[A followup on two related posts: Area of a surface confined by a sphere Area of a elliptic surface confined by a sphere . Thanks to all the inputs so far.] Let $S$ be a surface enclosed inside the ...
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Area of a elliptic surface confined by a sphere

Let $S$ be a surface enclosed inside the unit sphere in $R^3$. If every point of S is elliptic, then must $\operatorname{Area}(S)≤\operatorname{Area}(S^2)$?
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Area of a surface confined by a sphere

Let $S$ be a hypersurface enclosed inside the unit sphere in $R^n$. We may assume that every ray $\{t x: t \geq 0 \}$ intersects $S$ at most once. Under what extra condition is ${\rm Area}(S) \leq {\...
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Is the radial projection map area increasing?

Let $S$ be a hypersurface enclosed inside the unit sphere in $R^n$. Assume that every ray $\{t x: t \geq 0 \}$ intersects $S$ at most once. Is it always true that ${\rm Area}(S) \leq {\rm Area}(P(S))$...
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1 answer
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Compute lower bound on $\min_{E} \mathcal N(0,\sigma^2 I_n)(E)$ subject to $vol(E \cap H_n(r)) / vol(H_n(r)) \ge p$ where $H_n(r)$ is $n$-hemisphere

Let $n \ge 2$ be an integer, which may be assumed to be very large. For $r > 0$, consider the hemi-sphere $H_n(r) := S_n(r) \cap (\mathbb R^+ \times \mathbb R^{n-1})$, where $$ S_n(r):= \{x \in \...
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Estimate of volume of a ball on the boundary of Riemannian manifold

Let $M^n$ be a smooth compact Riemannian manifold with geodesically locally convex boundary and sectional curvature at least $-1$. Let $x\in M$ and $\varepsilon\in (0,1)$. Does there exist a ...
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4 votes
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Area lower bound given a mean curvature upper bound?

If $\Sigma$ is a smooth embedded closed hypersurface in $\mathbb R^n$ with (normalized) mean curvature $H\le 1$ (the mean curvature of the unit sphere), then its ($(n-1)$-dimensional) area is at least ...
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Isoperimetric profile with obstacle

Fix two open smooth bounded domains $\Omega_-$ and $\Omega_+$ with $\overline{\Omega}_-\subset \Omega_+$ in a complete Riemannian manifold ($\mathbb{R}^n$ is already interesting to me). I was ...
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3 votes
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Isoperimetry on $[0, 1]^n$ w.r.t $\ell_p$ distance, with $p \in [1,\infty]$

Let $A$ be a measurable subset of the metric space $\mathcal X = ([0, 1]^n,\ell_p)$ with $1 \le p \le \infty$, and define its $\varepsilon$-blowup by $A^\varepsilon:=\{x \in \mathcal X \mid \|x-a\|_p \...
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8 votes
1 answer
313 views

An isoperimetric-type inequality inside a cube

I am looking for a reference for the following inequality: if $\Omega \subset [0,1]^d$ satisfies $\mbox{vol}(\Omega) \leq 1/2$, then $$ \mathcal{H}^{d-1}\left( \partial\Omega \cap (0,1)^d\right) \geq ...
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3 votes
0 answers
131 views

What to do when Euler Lagrange Equation is highly nonlinear ode?

In $\mathbb{R}^3$, suppose there is a curve on X-Y plane $y(x)$ defined on $x\in [-a,a]$ satisfying: $y(x)\geqslant 0$; $y(-a)=y(a)=0.$ Rotate $y(x)$ along x-axis in $\mathbb{R}^3$ and get a solid ...
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3 votes
2 answers
271 views

What happens to the Gaussian volume of a Borel set when it is translated?

Let $\gamma_n$ be the standard Gaussian measure on $\mathbb R^n$, $A \subseteq \mathbb R^n$ be Borel and $c \in \mathbb R^n$. Define the translate $A_c := c + A := \{c+a \mid a \in A\} = \{x \in \...
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1 vote
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Gaussian isoperimetry for $\ell_p$ norms

Let $\gamma_n$ be the standard Gaussian measure on $\mathbb R^n$. It is well-known (e.g see Proposition 1) that for a given Gaussian volume content, half-spaces $H=\{x \in \mathbb R^n | a^Tx \le b\}$ ...
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1 vote
1 answer
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An asymptotic version of the Isoperimetric inequality

Let $U$ be a simply connected bounded open set in $\mathbb{R}^2$. The area of $U$ is denoted by $A$. (We do not assume any thing about its boundary). Assume that $\gamma_n$,s are smooth simple ...
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2 votes
0 answers
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Lower bound to $\epsilon$-expansion of a subset of a half-sphere

Below are two known lemmas on a $d$-dimensional sphere (related to the isoperimetric inequality). I would like to know: does a similar statement like this holds for a $d$-dimensional dome also (i.e. ...
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7 votes
2 answers
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Graph which do not satisfy a pseudo-Poincaré inequality

Say that an infinite (connected) graph (with vertices of bounded degree) satisfies a $\ell_1$-pseudo-Poincaré inequality if there is a constant $C>0$ so that for any $n \in \mathbb{N}$ for any ...
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10 votes
1 answer
376 views

Isoperimetric inequality for closed curves in $\mathbb{R}^n$

A well known isoperimetric inequality for closed curves in $\mathbb{R}^2$ can be generalized to closed curves in $\mathbb{R}^{2n}$, see: https://mathoverflow.net/a/321505/121665. I have two questions:...
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1 vote
1 answer
151 views

convergence and a mean curvature condition imply convexity

I have a question regarding the proof of theorem 4.6 in https://arxiv.org/abs/1007.3899 (Hall's conjecture). Let $S^2$ be the class of Borel subsets in $\mathbb{R}^2$ with finite and positive ...
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3 votes
1 answer
329 views

A question of Ahlswede and Katona: known lower bounds on $\beta(d,n)$?

Given a set $S\subseteq \{0,1\}^d$ of the Boolean hypercube of dimension $d$, define the average distance of $S$ as $$ \bar{d}(S) = \frac{1}{\lvert S\rvert^2} \sum_{x,y\in S} d_H(x,y)\tag{1} $$ where $...
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1 vote
1 answer
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gaussian isoperimetric result for minimal measure under translation

Consider two spherical Gaussian distributions in $\mathbb{R}^n$, $A = \mathcal{N}(x, I)$ and $B=\mathcal{N}(y, I)$ where the difference in means is $\delta = y - x$. Let $S \subset \mathbb{R}^n$ be a ...
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  • 33
8 votes
0 answers
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Hölder isoperimetric problem

Denote by $S_r$ the usual circle of radius $r$, with the path metric ($d(x,y) = r\theta$, where $\theta$ is the angle between the vectors $x$ and $y$), and let $\alpha \in (1/2,1)$. Consider the ...
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  • 507
3 votes
2 answers
111 views

area variation of a closed surface under ${\rm SL}(3)$

Let $\Sigma$ a closed oriented embedded surface in $R^3$. When $\Sigma$ is a round sphere, then for any smooth curve $A(t) \in SL(3)$ through the identity (i.e. $A(t) \in R^{3\times 3}$, $\det(A(t))=1$...
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  • 429
7 votes
2 answers
423 views

Visual proof of convergence for Steiner's symmetrization

I want to find a visual proof of the following fact: For any convex figure in the plane there is a sequence of Steiner's symmetrizations that makes it arbitrary close to a circular disc. All ...
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16 votes
0 answers
488 views

Isoperimetric inequality and geometric measure theory

The following version of the isoperimetric inequality can be easily deduced from the Brunn-Minkowski inequality: Theorem. If $K\subset\mathbb{R}^n$ is compact, then $$ |K|^{\frac{n-1}{n}}\leq n^{-1}...
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2 votes
0 answers
67 views

A uniform version of Minkowski content?

Let $A\subseteq [-1,1]^d$ be a measurable set and $\mu$ be the Lebesgue measure. For any $\delta>0$, define $A_\delta := \{x: d(x,A)\leq \delta\}$, where $d(x,A) := \inf_{y\in A}\|x-y\|_2$. The ...
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7 votes
1 answer
383 views

An isoperimetric type of inequality in terms of Wasserstein distance/Optimal transport

Let $A \subset \mathbb{R}^n$ be a region having the same volume as an $n$ dimensional ball $B^n_R$ with radius $R$ centring at the origin. Isoperimetric inequality says: $ Vol_{n-1} \partial A \geq ...
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4 votes
0 answers
46 views

Lower estimate on length of boundary of 2d Riemannian surface

Fix constants $\kappa\in \mathbb{R}, D>0,A>0$. Does there exist a constant $C>0$ depending on $\kappa, D, A$ only such that for any compact 2-dimensional Riemannian surface (or more generally ...
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  • 19k
8 votes
1 answer
247 views

Isoperimetric inequality on the plane

Let $A$ be a connected compact domain with smooth boundary in the Euclidean 2-plane. Assume its diameter is at most $d$. Assume that the second fundamental form of the boundary is at most $-c$ where $...
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