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3
votes
1answer
119 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
191 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 ...
2
votes
0answers
218 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
398 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 ...
20
votes
3answers
651 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
130 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
136 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 ...
4
votes
3answers
220 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 ...
2
votes
1answer
117 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 ...
8
votes
1answer
336 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 ...
1
vote
1answer
228 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 ...
6
votes
1answer
482 views

Isoperimetry and Poincare Inequality

What are the known relations between isoperimetric and Poincare inequalities on manifolds? For example, for manifolds with a lower bound on Ricci curvature, the Cheeger-Buser inequality relates the ...
6
votes
4answers
404 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 ...
1
vote
0answers
139 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
1answer
303 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 ...
12
votes
1answer
384 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 ...
3
votes
0answers
144 views

An isoperimetric inequality for “order” polytopes

I am looking for an isoperimetric inequality for order-like polytopes. An order polytope $K\in \mathbb{R}^n$ is defined by a set of linear inequaities: $$ \forall i \; 0\leq x_i \leq 1 $$ and $ ...
7
votes
2answers
470 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 ...
12
votes
2answers
632 views

Isoperimetric-like inequality for non-convex 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 ...
12
votes
1answer
634 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 ...
14
votes
3answers
839 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 $r_{in}$ ...
8
votes
2answers
986 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 ...