In orbifolds, small isoperimetric regions are small balls
Frank Morgan
在一个紧凑的orbifold中，对于小的规定体积，一个等周区域接近一个小的度量球;在一个欧几里德orbifold中，它是一个小的度量球。
In a compact orbifold, for small prescribed volume, an isoperimetric region is close to a small metric ball; in a Euclidean orbifold, it is a small metric ball.
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A local isoperimetric inequality for balls with nonpositive curvature
本文研究了球体在非正曲率度量下的局部等周不等式，证明小扰动不影响等周比，并揭示了保持等周比的条件是相似变换。
We have discovered a "little" gap in our proof of the sharp conjecture that in $\mathbb{R}^n$ with volume and perimeter densities $r^m$ and $r^k$, balls about the origin are uniquely isoperimetric if $0 < m \leq k - k/(n+k-1)$, that is, if they are stable (and $m > 0$). The implicit unjustified assumption is that the generating curve is convex.
We prove that on an essentially non-branching $\mathrm{MCP}(K,N)$ space, if a geodesic ball has a volume lower bound and satisfies some additional geometric conditions, then in a smaller geodesic ball (in a quantified sense) we have an estimate on the isoperimetric constants.
We consider the class of $λ$-concave bodies in $\mathbb R^{n+1}$; that is, convex bodies with the property that each of their boundary points supports a tangent ball of radius $1/λ$ that lies locally (around the boundary point) inside the body. In this class we solve a reverse isoperimetric problem: we show that the convex hull of two balls of radius $1/λ$ (a sausage body) is a unique volume minimizer among all $λ$-concave bodies of given surface area. This is in a surprising contrast to the standard isoperimetric problem for which, as it is well-known, the unique maximizer is a ball. We solve…
In this paper we solve several reverse isoperimetric problems in the class of $λ$-convex bodies, i.e., convex bodies whose curvature at each point of their boundary is bounded below by some $λ> 0$. We give an affirmative answer in $\mathbb{R}^3$ to a conjecture due to Borisenko which states that the $λ$-convex lens, i.e., the intersection of two balls of radius $1/λ$, is the unique minimizer of volume among all $λ$-convex bodies of given surface area. Also, we prove a reverse inradius inequality: in model spaces of constant curvature and arbitrary dimension, we show that the $λ$-convex lens (p…
Motivated by Perelman's Pseudo Locality Theorem for the Ricci flow, we prove that if a Riemannian manifold has Ricci curvature bounded below in a metric ball which moreover has almost maximal volume, then in a smaller ball (in a quantified sense) it holds an almost-euclidean isoperimetric inequality. The result is actually established in the more general framework of non-smooth spaces satisfying local Ricci curvature lower bounds in a synthetic sense via optimal transportation.
The minimizers of the anisotropic fractional isoperimetric inequality with respect to the convex body $K$ in $\mathbb{R}^n$ are shown to be equivalent to star bodies whenever $K$ is strictly convex and unconditional. From this a Pólya-Szegö principle for anisotropic fractional seminorms is derived by using symmetrization with respect to star bodies.
Klartag's needle decomposition technique enables one to obtain strong isoperimetric inequalities on Riemannian manifolds other than the classical known examples. As a result, in this paper, we obtain sharp isoperimetric inequalities for compact rank one symmetric spaces (CROSS). Namely, for the real projective space $\mathbb{R}P^n$, we demonstrate that the isoperimetric regions are given by either the geodesic balls or tubes around some $\mathbb{R}P^k\subset\mathbb{R}P^n$. For the complex projective space $\mathbb{C}P^n$, the isoperimetric regions are given by either the geodesic balls or tube…
An isoperimetric inequality on the $\ell_p$ balls
$\ell_p^n$单位球上的归一化体积测度（$1\leq p\leq 2$）满足以下等周不等式：一组测度$a$的边界测度至少为$cn^{1/p}\tilde{a}\log^{1-1/p}(1/\tilde{a})$，其中$\tilde{a}=\min(a,1-a)$ .
Asymptotic isoperimetry of balls in metric measure spaces
本文研究度量测度空间中球体积的渐近性态。我们首先介绍了一个一般的设置适应于研究的渐近等周在一般类的度量测度空间。然后，我们引入了一个概念的“渐近等周”的家庭有限和无界的措施子集。证明了存在多项式一致增长的图，其球不是渐近等周的，并讨论了相关性质在拟等周下的稳定性。最后，我们研究了度量测度空间中连通子集的渐近等周性质。特别是，我们建立了统一的多项式增长的连通子集不是渐近等周图。
Isoperimetric problem and structure at infinity on Alexandrov spaces with nonnegative curvature
研究非负曲率有限维Alexandrov空间的等周剖面、等周集存在性与无穷远渐近结构的关系，证明线性体积增长等条件的等价性，并给出2维情形下等周集的存在性与拓扑刻画。
Topological regularity of isoperimetric sets in PI spaces having a deformation property
本文证明了具有形变性质的PI空间中等周集的拓扑正则性，揭示了其开集、边界密度估计及有界性，并推广至塌缩RCD空间，简化了经典欧几里得论证。
We prove an inequality, valid on any finitely generated group with a fixed finite symmetric generating set, involving the growth of successive balls, and the average length of an element in a ball. It generalizes recent improvements of the Coulhon Saloff-Coste inequality. We reformulate the inequality in terms of the Følner function; in the case the finitely generated group is amenable with exponential growth, this allows us to express the best possible (outer) constant in the Coulhon Saloff-Coste isoperimetric inequality with the help of a formula involving the growth rate and the asymptotic …
Tropical measures, anisotropic isoperimetric inequality and honeycomb
本文引入了基于热带度量的热带球面测度，并证明了二维中的精确热带等周不等式，推导出热带蜂窝定理，同时探讨了热带Minkowski内容和相关几何结构。
Some inequalities of isoperimetric type for the c-affine surface area
针对Schütt等人新引入的c-仿射表面积Ω^c，证明球体类中其由半径n/(n+1)的球最大化，建立Santaló型不等式，并得到涉及表面积的更复杂不等式。
We establish a structure theorem for minimizing sequences for the isoperimetric problem on noncompact $\mathsf{RCD}(K,N)$ spaces $(X,\mathsf{d},\mathcal{H}^N)$. Under the sole (necessary) assumption that the measure of unit balls is uniformly bounded away from zero, we prove that the limit of such a sequence is identified by a finite collection of isoperimetric regions possibly contained in pointed Gromov--Hausdorff limits of the ambient space $X$ along diverging sequences of points. The number of such regions is bounded linearly in terms of the measure of the minimizing sequence. The result f…
Two properties of optimisers for the reverse isoperimetric problem
研究黎曼流形中在给定半径球内自由滚动且固定体积的凸集，证明无C^2光滑的周长最大化者，并证明C^2光滑区域的最小主曲率为常数。
We prove a randomized version of the generalized Urysohn inequality relating mean-width to the other intrinsic volumes. To do this, we introduce a stochastic approximation procedure that sees each convex body K as the limit of intersections of Euclidean balls of large radii and centered at randomly chosen points. The proof depends on a new isoperimetric inequality for the intrinsic volumes of such intersections. If the centers are i.i.d. and sampled according to a bounded continuous distribution, then the extremizing measure is uniform on a Euclidean ball. If one additionally assumes that the …
For a complete noncompact connected Riemannian manifold with bounded geometry $M^n$, we prove that the isoperimetric profile function $I_{M^n}$ is a locally $\left(1-\frac{1}{n}\right)$-Hölder continuous function and so in particular it is continuous. Here for bounded geometry we mean that $M$ have $Ricci$ curvature bounded below and volume of balls of radius $1$, uniformly bounded below with respect to its centers. We prove also the equivalence of the weak and strong formulation of the isoperimetric profile function in complete Riemannian manifolds which is based on a lemma having its own int…
In this paper we study regularity and topological properties of volume constrained minimizers of quasi-perimeters in $\sf RCD$ spaces where the reference measure is the Hausdorff measure. A quasi-perimeter is a functional given by the sum of the usual perimeter and of a suitable continuous term. In particular, isoperimetric sets are a particular case of our study. We prove that on an ${\sf RCD}(K,N)$ space $({\rm X},{\sf d},\mathcal{H}^N)$, with $K\in\mathbb R$, $N\geq 2$, and a uniform bound from below on the volume of unit balls, volume constrained minimizers of quasi-perimeters are open bou…
The isoperimetric problem for four-dimensional parallelohedra
本文证明正则24胞腔在固定体积的四维平行多面体中唯一最小化表面积，并建立二次Hausdorff稳定性及二阶变分公式，识别严格局部极小值的根格Voronoi胞腔。
A note on Laplacian bounds, deformation properties and isoperimetric sets in metric measure spaces
本文构造无开代表元等周集的长度 PI 空间反例，并证明距离平方函数的上 Laplacian 界可保证形变性质，从而推出本质非分支 MCP/CD 空间中等周集具有开且在体积条件下有界的代表元。
The Euclidean concentration inequality states that, among sets with fixed volume, balls have $r$-neighborhoods of minimal volume for every $r>0$. On an arbitrary set, the deviation of this volume growth from that of a ball is shown to control the square of the volume of the symmetric difference between the set and a ball. This sharp result is strictly related to the physically significant problem of understanding near maximizers in the Riesz rearrangement inequality with a strictly decreasing radially decreasing kernel. Moreover, it implies as a particular case the sharp quantitative Euclidean…
Volume entropy, systole and stable norm on graphs
研究了图的一些新的等周不等式。本文建立了体积熵（或渐近体积）、收缩与加权图的第一个贝蒂数之间的关系。我们还找到了一些特殊的措施，单位球的稳定规范的图的体积的界限。
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