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Questions tagged [isoperimetric-problems]

3
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2answers
151 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 \...
1
vote
0answers
61 views

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 $H=\{x \in \mathbb R^n | a^Tx \le b\}$ ...
1
vote
1answer
96 views

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 ...
2
votes
0answers
23 views

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. ...
5
votes
2answers
255 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 ...
10
votes
1answer
240 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:...
2
votes
1answer
91 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 ...
3
votes
1answer
246 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 $...
1
vote
1answer
61 views

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 ...
7
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0answers
76 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 ...
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0answers
104 views

Isoperimetric inequality inside a regular polygon

Let $\mathcal{P}_n$ be a fixed $n$-sided regular polygon with area $A:=\vert \mathcal{P}_n\vert>0$. For any $c\in (0,A)$, I would like to find the shape of the domain $D\subset \mathcal{P}_n$ such ...
3
votes
2answers
101 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. $A(t) \in R^{3\times 3}$, $\det(A(t))=1$...
7
votes
2answers
248 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 ...
11
votes
0answers
278 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}...
2
votes
0answers
55 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 ...
7
votes
1answer
311 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 ...
4
votes
0answers
41 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 ...
8
votes
1answer
229 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 $...
8
votes
1answer
185 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 ...
4
votes
1answer
379 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 ...
3
votes
1answer
131 views

Work on “Churning Polygons”

Background of this question is that I recently stumbled over the problem of deforming polygons in area-preserving way, i.e. modifying the angles between adjacent edges while preserving edge-lengths, ...
5
votes
1answer
193 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 ...
2
votes
1answer
907 views

Shannon's proof of the entropy power inequality

In Shannon's paper on information theory, found here, he asserts the entropy power inequality in appendix 6, found on page 52. I was reading his proof and it seems like there is a gap. Through his ...
30
votes
1answer
1k 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 ...
0
votes
2answers
81 views

Level sets and integral of functions of two variables

Let $f_1,f_2$ be two positive functions on $\Omega_1, \Omega_2 \subset R^2$ with $f_1|_{\partial \Omega_1}=f_2|_{\partial \Omega_2}=0$. For every $\lambda>0$, denote the the area of the domain ...
7
votes
2answers
329 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) ...
6
votes
0answers
160 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 ...
11
votes
0answers
255 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 ...
8
votes
3answers
269 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 ...
5
votes
1answer
249 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 ...
20
votes
2answers
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 ...
5
votes
0answers
147 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 ...
3
votes
1answer
133 views

Are there isomeasure simplices?

Say that two polyhedra in $\mathbb{R}^3$ have isomeasures (my terminology) if they have: the same volume, the same surface area, the same sum of all edge lengths, and the same number of vertices. The ...
2
votes
1answer
227 views

Isocapacity inequalities in the theory of Sobolev Spaces

Section 9 of the lectures notes of Maz'ya (Мазья) on isocapacity, lectures notes that can be found at: http://www.math.liu.se/~vlmaz/pdf/mazya.pdf, discusses inequality between the $L^p$ norm in a ...
9
votes
1answer
511 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}{\...
6
votes
1answer
515 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 ...
4
votes
2answers
644 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 ...
23
votes
3answers
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 ...
1
vote
1answer
278 views

Peculiar vertex-isoperimetric inequality on the discrete torus (and generalization)

Consider a discrete even torus $G=(V,E)$, i.e. the graph on $\lbrace 0,1,\dots,n-1 \rbrace^2$, $n$ even, where two vertices are connected by an edge only if they differ by 1 in only one coordinate, ...
0
votes
0answers
266 views

Isoperimetric profile

In the paper of Andrews and Bryan Curvature bounds by isoperimetric comparison for normalized Ricci flow on the two-sphere http://arxiv.org/abs/0908.3606, the isoperimetric profile is defined by $h(\...
5
votes
3answers
275 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 ...
3
votes
1answer
137 views

Can the isoperimetric dimension of a d-generated group attain any value?

Background The isoperimetric dimension of a finitely generated group $G$, which we denote by $\dim(G)$, is the largest number $d$ such that any Cayley graph $\Gamma$ of $G$ (constructed with respect ...
9
votes
1answer
509 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-|...
1
vote
1answer
369 views

Hypercube isoperimetric inequality for non-increasing events

It is well known that isoperimetric inequalities on a hypercube are closely related to influences, but all the theorems I'm aware of deal with monotone sets. Now suppose we have an arbitrary set $X \...
8
votes
1answer
1k 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 ...
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votes
4answers
550 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 ...
2
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0answers
183 views

Bonnesen's inequality for non-simple curves

Given a closed curve in the plane $\mathbb{R}^2$, it is well known that $L^2 \geq 4\pi A$ where $L$ is the length of the curve and $A$ is the area of the interior of the curve. For a simple closed ...
4
votes
2answers
1k 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 $\...