Questions tagged [isoperimetric-problems]
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Improvement of isoperimetric inequalities
The standard functional isoperimetric inequality is for an integer $n\ge 1$,
$$
\Vert u\Vert_{L^{\frac{n}{n-1}}(\mathbb R^n)}\le c(n)\Vert \nabla u\Vert_{L^1(\mathbb R^n)}, \quad c(n)=\frac{(\vert\...
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Sub-additiviy of the log-Sobolev constant without independence
If two random variables $X$ and $Y$ verify the log-Sobolev inequality, what can we say about the log-Sobolev constant of their sum $X+Y$?
If they are independent, we know that
$$
c_{LS}(X+Y) \leq c_{...
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Isoperimetric Inequalities in Annular Regions
Let $\Omega$ be an open set in $\mathbb{R}^2$ whose boundary is a rectifiable Jordan curve. Then an old result by Alfred Huber states that
$$
\left(\int_{\partial \Omega} e^u ds\right)^2 \geq 2 \left(...
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Symmetry of the isoperimetric profile
Given a probability measure $\mu$ on a metric space $(X, \mathsf{d})$, the $(\mu-)$Minkowski content of a set $A$ is defined as
$$\mu^+ (A) := {\lim\inf}_{r \to 0^+} \frac{\mu ( A_r \setminus A)}{r},$$...
- 706
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Cheeger constant and isoperimetric ratio
$(S^2,g)$ is 2-dimensional sphere with Riemannian metric. Consider any curves $\gamma$ on $S^2$ dividing the total area $A$ into two parts $A_1+A_2 =A$. The isoperimetric ratio is
$$
C_s(\gamma)=\frac{...
- 165
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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 ...
- 91
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Reference for Varopoulos isoperimetric inequality with multiplicity
The Varopoulos isoperimetric inequality for a bounded domain $D$ in a nilpotent group $\Gamma$ of growth $n$ reads
$$
\# D \le \mathrm{const} \cdot (\#\partial D)^{n/(n-1)}
$$
See Ch. 6.E+ in Gromov's ...
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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 ...
- 4,362
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Dimension reduction and isoperimetric inequality
$\newcommand{\II}{\mathit{II}}$The isoperimetric inequality $\II_n$ in ${\mathbb R}^n$ is
$$\frac{{\rm vol}_nU}{{\rm vol}_nB_n}\le\left(\frac{{\rm vol}_{n-1}\partial U}{{\rm vol}_{n-1}\partial B_n}\...
- 51.9k
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Sphere with bounded curvature
Let $V$ be a body in $\mathbb{R}^3$ bounded by a smooth sphere with principal curvatures at most 1 (by absolute value).
Is it true that
$$\mathop{\rm vol} V\ge \mathop{\rm vol} B,$$
where $B$ denotes ...
- 44.4k
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Nonlocal perimeter of level sets
Let $u \in W^{s,1}(B)$ be given and $k < l$ be two numbers, then I am looking for a way to bound the following term from above. Here $B$ is the euclidean ball.
$$
\int_{B: u < k} \int_{B:u>l} ...
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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 ...
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$\newcommand\v{\operatorname{vol}_d(C}$Compact subsets of $ℝ^d$ which maximize $\inf_{|v|\le1}\dfrac{\v\cap(𝜀v+C))}{\v)}$ for fixed $\v)$ and $𝜀>0$
Let $\operatorname{vol}_d$ be the volume measure on $\mathbb R^d$ and let $B_d$ be the unit-ball. For $\varepsilon \ge 0$ and a compact subset $C$ of $\mathbb R^d$ with $\operatorname{vol}_d(C)>0$, ...
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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 $n$-...
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Isoperimetric inequality for general metric space
Consider some space $\mathcal{S}$ with metric $d$ and measure $\mu$.
For arbitrary set $H$ denote the $v$-bound of $H$ by $\delta_v(H):= \{x \mid x \notin H: \exists y \in H \text{ s.t. } d(x,y) \le v ...
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