Questions tagged [isoperimetric-problems]

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Why is Fourier analysis so handy for proving the isoperimetric inequality?

I have just completed an introductory course on analysis, and have been looking over my notes for the year. For me, although it was certainly not the most powerful or important theorem which we ...
Steven Gubkin's user avatar
31 votes
1 answer
2k views

A long-lasting conjecture of Pólya & Szegő

There is a conjecture by Pólya & Szegő (~1950, stated in p. 159 of their book Isoperimetric Inequalties in Mathematical Physics) which is as follows: "Of all $n$-gons of a fixed area, the regular ...
BigM's user avatar
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24 votes
3 answers
1k views

Isoperimetric inequality on a Riemannian sphere

Consider a two-dimensional sphere with a Riemannian metric of total area $4\pi$. Does there exist a subset whose area equals $2\pi$ and whose boundary has length no greater than $2\pi$? (To avoid ...
Sergei Ivanov's user avatar
22 votes
0 answers
502 views

Sphere with bounded curvature

Let $V$ be a body in $\mathbb{R}^3$ bounded by a smooth sphere with principal curvatures at most 1 (by absolute value). Is it true that $$\mathop{\rm vol} V\ge \mathop{\rm vol} B,$$ where $B$ denotes ...
Anton Petrunin's user avatar
20 votes
2 answers
2k views

"a shape that ... lies halfway between a square and a circle"

An article in the Notices of the AMS, Volume 61, Issue 10, 2014 (PDF download link), on Khot's Unique Games Conjecture, says this: Another group ... found a shape that in a certain sense lies ...
Joseph O'Rourke's user avatar
17 votes
2 answers
945 views

Isoperimetric-like inequality for non-connected sets

The classical isoperimetric inequality can be stated as follows: if $A$ and $B$ are sets in the plane with the same area, and if $B$ is a disk, then the perimeter of $A$ is larger than the perimeter ...
Guillaume Aubrun's user avatar
16 votes
1 answer
1k views

Optimal 8-vertex isoperimetric polyhedron?

I know from Marcel Berger's Geometry Revealed: A Jacob's Ladder to Modern Higher Geometry (p.531) that it is not yet established which polyhedron in $\mathbb{R}^3$ on 8 vertices achieves the optimal ...
Joseph O'Rourke's user avatar
16 votes
0 answers
588 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}...
Piotr Hajlasz's user avatar
15 votes
3 answers
1k views

Stronger version of the isoperimetric inequality

I have been searching for a version of the isoperimetric inequality which is something like: $P(\Omega) - 2\sqrt{\pi} Vol(\Omega)^{1/2} \geq \pi (r_{out}^2 - r_{in}^2)$ where $r_{out}$ and $r_{in}$ ...
Dorian's user avatar
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12 votes
1 answer
656 views

General Isoperimetric Inequality via Representation Theory of SO(n)

Is there a known proof of the $n$-dimensional isoperimetric inequality which generalizes Hurwitz's proof using Fourier analysis in the $2$-dimensional case? Specifically, I imagine such a proof would ...
Sean Eberhard's user avatar
12 votes
0 answers
349 views

A variation on the local Günther inequality

This question is about a variation on the Günther (also known as Günther-Bishop) inequality for manifolds of sectional curvature bounded from above. With Greg Kuperberg, we would deduce from it a ...
Benoît Kloeckner's user avatar
11 votes
2 answers
2k views

Levy's isoperimetric inequality for sphere

Let me recall subj: If $s>0$, $A$ and $B$ are two subsets of $\mathbb{S}^{n}$, $|A|=|B|$ ($|\cdot|$ stands for the Lebesgue measure on the sphere) and $B$ is a cup $B=\{ (x_1,x_2,\dots,x_n)\in \...
Fedor Petrov's user avatar
11 votes
2 answers
739 views

Isoperimetric inequality in complex hyperbolic space

Let $\mathbb{H}_\mathbb{C}^n$ be n-dimensional complex hyperbolic space. This space is a complex analog of hyperbolic space. It is isometric to the quotient of hyperboloid $$|z_0|^2-|z_1|^2-\dots-|...
ε-δ's user avatar
  • 1,785
10 votes
4 answers
620 views

Packing and isoperimetrics

Suppose a manufacturer bottles small units of liquid and ships them via very large trucks. If the transportation cost nothing, spherical bottles would minimize the packaging cost (isoperimetric ...
David Feldman's user avatar
10 votes
2 answers
886 views

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 $D \subseteq \mathbb{R}^2$...
Asaf Shachar's user avatar
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10 votes
1 answer
445 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: ...
Piotr Hajlasz's user avatar
9 votes
1 answer
2k views

Isoperimetry and Poincaré Inequality

What are the known relations between isoperimetric and Poincaré inequalities on manifolds? For example, for manifolds with a lower bound on Ricci curvature, the Cheeger-Buser inequality relates the ...
ThiKu's user avatar
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9 votes
1 answer
544 views

What is the shape of the $n$-gon which gives the maximum of a function?

What is the shape of the $n$-gon $P_1P_2\cdots P_n$ which gives the maximum of $A_n$? The quantity $A_n$ is defined by $$ A_n = \frac{{\sum_{i\lt{j}\le{n}}{\lvert P_i P_j\rvert}^2}-{\sum_{i=1}^{n}{\...
mathlove's user avatar
  • 4,727
8 votes
3 answers
285 views

Set with small internal radius, small perimeter and prescribed area

Given a regular set $E\subset \mathbb R^2$ define $$ R(E) = \sup\{r\colon \exists x,\ B(x,r)\subseteq E\} $$ to be the radius of the largest circle contained in $E$ and let $|\partial E|$ be the ...
Emanuele Paolini's user avatar
8 votes
1 answer
263 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 $...
asv's user avatar
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8 votes
2 answers
779 views

A hypercube-related graph

For integer $n\ge 3$, consider the graph on the set of all even vertices of the $n$-dimensional hypercube $\{0,1\}^n$ in which two vertices are adjacent whenever they differ in exactly two coordinates....
Seva's user avatar
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8 votes
1 answer
438 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 ...
Stefan Steinerberger's user avatar
8 votes
1 answer
234 views

Poincare's argument for maximizing the Coulomb energy

For $\Omega\subset \mathbb{R}^3$ a region with $|\Omega| = |B_1|$, let $$ C(\Omega) = \int_\Omega\int_\Omega \frac{dxdy}{|x-y|} $$ denote the Coulomb (or gravitational, etc) energy. Poincaré is ...
Otis Chodosh's user avatar
  • 7,077
8 votes
1 answer
318 views

Mass transportation proof of the Gaussian isoperimetric inequality?

In his book "Topics in optimal transportation", Graduate Studies in Mathematics 58, AMS 2003, Villani presents a proof, due to Gromov, of the classical isoperimetric inequality in Euclidean ...
Xiazhong Zhu's user avatar
8 votes
0 answers
109 views

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 ...
mdr's user avatar
  • 527
7 votes
2 answers
356 views

Closed curve whose neighborhood is as large as possible

Let $C$ be a closed curve in the plane and let $N_\epsilon(C)$ be an $\epsilon$-neighborhood of $C$, like this: (ignore the fact that the "curve" is polygonal in this picture, I drew it in MATLAB) ...
Tom Solberg's user avatar
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7 votes
2 answers
404 views

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 ...
ARG's user avatar
  • 4,342
7 votes
2 answers
506 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 ...
Anton Petrunin's user avatar
7 votes
3 answers
613 views

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 ...
Beni Bogosel's user avatar
  • 2,102
7 votes
1 answer
207 views

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($\gamma$)$=L$ }\}....
April's user avatar
  • 389
7 votes
1 answer
281 views

Convexity of Isoperimetric Domains

I am interested in what is known about the convexity of isoperimetric domains in compact Cartan-Hadamard manifolds (Riemannian manifolds that are complete and simply-connected and have non-positive ...
Alec Payne's user avatar
7 votes
1 answer
424 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 ...
random_shape's user avatar
7 votes
0 answers
648 views

Least area minimal hypersurface of $\mathbb C P^{n+1}$

After a few lectures on min-max for minimal hypersurfaces and isoperimetric problems, and seeing in several instances that the least area minimal hypersurface of the round sphere is an equator, I was ...
Renato G. Bettiol's user avatar
6 votes
2 answers
424 views

$L^{p}$ isoperimetric inequalities on the Hamming cube

Let $A \subset \{-1,1\}^{n}$ be a subset of the Hamming cube with cardinality $|A|=2^{n-1}$. Define $w_{A} : \{-1,1\}^{n} \to \mathbb{N}\cup \{0\}$ so that $w_{A}(x)$ to be number of boundary edges ...
Paata Ivanishvili's user avatar
6 votes
0 answers
176 views

Optimal planar net for catching convex shapes

Imagine you want to make a net out of string to filter and catch objects of a certain size, minimizing the length of string employed. (This actually arises in filtering biological impurities from ...
Joseph O'Rourke's user avatar
5 votes
3 answers
349 views

Perimeter/Neighborhood of a graph on grid

Hello, I have a $\sqrt{n}\times\sqrt{n}$ lattice graph $G=(V,E)$ i.e. vertices on said 2-dim integer lattice, and two vertices have an edge if their $L_1$ distance is one. Now I want to claim ...
Mudi's user avatar
  • 93
5 votes
1 answer
183 views

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 $G = (\Omega, E)$ ...
Sam OT's user avatar
  • 560
5 votes
1 answer
174 views

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 $\operatorname{Area}(S)≤\operatorname{Area}(S^2)$?
Thomas's user avatar
  • 511
5 votes
1 answer
303 views

Panning for gold nuggets: a type of isoperimetric problem

Let $C$ be a unit-radius circle in the plane. Suppose you have a total length $L$ of string available, and your task is to connect chords of $C$ using no more than $L$ of string to minimize the ...
Joseph O'Rourke's user avatar
5 votes
1 answer
627 views

An inequality inspired by the isoperimetric inequality

Let us consider the simplest isoperimetric inequality. Consider a smooth simple closed curve given by $r=\rho(\theta)$ in polar coordinates, where $\rho(\theta)>0$ can be regarded as a smooth ...
user50396's user avatar
  • 141
5 votes
1 answer
175 views

"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. ...
sobol's user avatar
  • 151
5 votes
0 answers
72 views

Dimension reduction and isoperimetric inequality

$\newcommand{\II}{\mathit{II}}$The isoperimetric inequality $\II_n$ in ${\mathbb R}^n$ is $$\frac{{\rm vol}_nU}{{\rm vol}_nB_n}\le\left(\frac{{\rm vol}_{n-1}\partial U}{{\rm vol}_{n-1}\partial B_n}\...
Denis Serre's user avatar
  • 51.5k
5 votes
0 answers
84 views

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 ...
RBega2's user avatar
  • 2,423
4 votes
2 answers
2k views

Isoperimetric inequality in negative sectional curvature

Let $M$ be a complete, non-compact, simply connected Riemannian manifold of dimension $n$ whose sectional curvatures are bounded above by $\kappa<0$. I want to prove that for any open subset $\...
Marc's user avatar
  • 604
4 votes
2 answers
757 views

The Isoperimetric problem for domains constrained to lie between two parallel planes

It is well known that for a given volume $V$, a sphere is the shape that minimizes the surface area. I am interested in the same problem under the constraint that the shape must lie between the planes ...
Jules's user avatar
  • 463
4 votes
1 answer
115 views

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 $n$-...
user7868's user avatar
  • 620
4 votes
1 answer
661 views

Generalized widths and reverse Urysohn inequalities

This question is inspired by the discussion in MO questions "Local minimum from directional derivatives in the space of convex bodies" and "Bodies of constant width?" about generalized notions of ...
Yoav Kallus's user avatar
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4 votes
0 answers
140 views

A variation of Zuk's isoperimetric inequality for groups

$\DeclareMathOperator\diam{diam}\DeclareMathOperator\inrad{inrad}$There is a isoperimetric inequality (conjectured by Sikorav and proven by Żuk (Topology 39 (2000) 947–956) which holds in every Cayley ...
ARG's user avatar
  • 4,342
4 votes
0 answers
121 views

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 ...
Sarvesh Ravichandran Iyer's user avatar
4 votes
0 answers
126 views

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 ...
Nobody's user avatar
  • 41