# Questions tagged [isoperimetric-problems]

The tag has no usage guidance.

81 questions
Filter by
Sorted by
Tagged with
31 views

276 views

61 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 ...
365 views

207 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 ...
465 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 ...
140 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, ...
245 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 ...
1k 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 ...
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 ...
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 ...
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) ...