# Questions tagged [isoperimetric-problems]

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### Cheeger constant of truncated hypercube

Look at the $d$-dimensional hypercube and truncate it. This means one replaces each vertex by a cycle (of length $d$) in such a way the the new graph is 3-regular. Question 1: What is the asymptotic ...
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### Estimate of volume of a ball on the boundary of Riemannian manifold

Let $M^n$ be a smooth compact Riemannian manifold with geodesically locally convex boundary and sectional curvature at least $-1$. Let $x\in M$ and $\varepsilon\in (0,1)$. Does there exist a ...
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### Area lower bound given a mean curvature upper bound?

If $\Sigma$ is a smooth embedded closed hypersurface in $\mathbb R^n$ with (normalized) mean curvature $H\le 1$ (the mean curvature of the unit sphere), then its ($(n-1)$-dimensional) area is at least ...
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### Isoperimetric profile with obstacle

Fix two open smooth bounded domains $\Omega_-$ and $\Omega_+$ with $\overline{\Omega}_-\subset \Omega_+$ in a complete Riemannian manifold ($\mathbb{R}^n$ is already interesting to me). I was ...
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Let $A$ be a measurable subset of the metric space $\mathcal X = ([0, 1]^n,\ell_p)$ with $1 \le p \le \infty$, and define its $\varepsilon$-blowup by $A^\varepsilon:=\{x \in \mathcal X \mid \|x-a\|_p \... 1answer 253 views ### An isoperimetric-type inequality inside a cube I am looking for a reference for the following inequality: if$\Omega \subset [0,1]^d$satisfies$\mbox{vol}(\Omega) \leq 1/2$, then $$\mathcal{H}^{d-1}\left( \partial\Omega \cap (0,1)^d\right) \geq ... 0answers 91 views ### What to do when Euler Lagrange Equation is highly nonlinear ode? In \mathbb{R}^3, suppose there is a curve on X-Y plane y(x) defined on x\in [-a,a] satisfying: y(x)\geqslant 0; y(-a)=y(a)=0. Rotate y(x) along x-axis in \mathbb{R}^3 and get a solid ... 2answers 235 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 \... 0answers 71 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\} ... 1answer 100 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 ... 0answers 34 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. ... 2answers 290 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 ... 1answer 330 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:... 1answer 122 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 ... 1answer 296 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} $$where ... 1answer 77 views ### gaussian isoperimetric result for minimal measure under translation Consider two spherical Gaussian distributions in \mathbb{R}^n, A = \mathcal{N}(x, I) and B=\mathcal{N}(y, I) where the difference in means is \delta = y - x. Let S \subset \mathbb{R}^n be a ... 0answers 86 views ### Hölder isoperimetric problem Denote by S_r the usual circle of radius r, with the path metric (d(x,y) = r\theta, where \theta is the angle between the vectors x and y), and let \alpha \in (1/2,1). Consider the ... 0answers 125 views ### Isoperimetric inequality inside a regular polygon Let \mathcal{P}_n be a fixed n-sided regular polygon with area A:=\vert \mathcal{P}_n\vert>0. For any c\in (0,A), I would like to find the shape of the domain D\subset \mathcal{P}_n such ... 2answers 107 views ### area variation of a closed surface under {\rm SL}(3) Let \Sigma a closed oriented embedded surface in R^3. When \Sigma is a round sphere, then for any smooth curve A(t) \in SL(3) through the identity (i.e. A(t) \in R^{3\times 3}, \det(A(t))=1... 2answers 297 views ### Visual proof of convergence for Steiner's symmetrization I want to find a visual proof of the following fact: For any convex figure in the plane there is a sequence of Steiner's symmetrizations that makes it arbitrary close to a circular disc. All ... 0answers 382 views ### Isoperimetric inequality and geometric measure theory The following version of the isoperimetric inequality can be easily deduced from the Brunn-Minkowski inequality: Theorem. If K\subset\mathbb{R}^n is compact, then$$ |K|^{\frac{n-1}{n}}\leq n^{-1}... 0answers 60 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 ... 1answer 333 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 ...
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 ...