# Questions tagged [isoperimetric-problems]

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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 $$\mu^+ (A) := {\lim\inf}_{r \to 0^+} \frac{\mu ( A_r \setminus A)}{r},$$...
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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 ...
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### Isoperimetric type inequality in $\mathbb{R}^2$

Fix $L \in (0,\infty)$ and consider $\mathcal{C}_L$ defined as follows: \begin{align*} \mathcal{C}_L := \{ \gamma:[0,1] \rightarrow \mathbb{R}^2 |~ \gamma \text{ is smooth and length($\gamma$)$=L$ }\}....
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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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### 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 ...
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### 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 ...
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### 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. $A(t) \in R^{3\times 3}$, $\det(A(t))=1$...