The isoperimetry tag has no wiki summary.

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### “Isoperimetric” inequality for Martin capacity [on hold]

After some fruitless literature search, I was wondering if anyone has proved an "isoperimetric" inequality for Martin capacities (for transient Brownian motion with $d\geq 3$):
...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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, ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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
...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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
$ ...

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### 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 ...

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### 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 ...

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### 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 $r_{in}$ ...

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### 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 ...