Wenxiong Chen, Ralph Howard, Erwin Lutwak, Deane Yang, Gaoyong Zhang
A purely analytic proof is given for an inequality that has as a direct consequence the two most important affine isoperimetric inequalities of plane convex geometry: The Blaschke-Santalo inequality and the affine isoperimetric inequality of affine differential geometry.
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A stability version of the reverse isoperimetric inequality, and the corresponding inequality for isotropic measures are established.
We give a refinement of the quantitative isoperimetric inequality. We prove that the isoperimetric gap controls not only the Fraenkel asymmetry but also the oscillation of the boundary.
The isodiametric inequality is derived from the isoperimetric inequality trough a variational principle, establishing that balls maximize the perimeter among convex sets with fixed diameter. This principle brings also quantitative improvements to the isodiametric inequality, shown to be sharp by explicit nearly optimal sets.
An affine isoperimetric inequality for log-concave functions
本文提出针对对数凹函数的泛函等周不等式，推广了 Lutwak、Yang 和 Zhang 的仿射等周不等式，并给出了极体体积的下界。
An isoperimetric inequality for mean shadow
研究在单位面积平面Borel集类中最小化Favard长度的问题，证明圆是最优解。
M. Gromov introduced the Lipschitz order relation on the set of metric measure spaces and developed a rich theory. In particular, he claimed that an isoperimetric inequality on a non-discrete space is represented by using the Lipschitz order. We relax the definition of the Lipschitz order allowing an additive error to relate with an isoperimetric inequality on a discrete space. As an application, we obtain an isoperimetric inequality on the non-discrete $n$-dimensional $l^1$-cube by taking the limits of an isoperimetric inequality of the discrete $l^1$-cubes.
We prove new $L_p$ affine isoperimetric inequalities for all $ p \in [-\infty,1)$. We establish, for all $p\neq -n$, a duality formula which shows that $L_p$ affine surface area of a convex body $K$ equals $L_\frac{n^2}{p}$ affine surface area of the polar body $K^\circ$.
In this paper, we develop a basic theory of Orlicz affine and geominimal surface areas for convex and $s$-concave functions. We prove some basic properties for these newly introduced functional affine invariants and establish related functional affine isoperimetric inequalities as well as functional Santaló type inequalities.
We improve the isoperimetric inequality of Coulhon and Saloff-Coste following a method of mass transport proposed by Gromov.
We observe after Bayle and Rosales that the Levy-Gromov isoperimetric inequality generalizes to convex manifolds with boundary.
Affine isoperimetric inequalities on flag manifolds
本文基于Furstenberg和Tzkoni的工作，在旗流形上引入r-旗仿射拟质量积分及其对偶版本，推广相关拟质量积分，建立不变性并扩展基本结果，证明凸几何中的仿射等周不等式及逆形式，还引入函数形式并建立对应不等式。
A stability version of the Blaschke-Santaló inequality and the affine isoperimetric inequality for convex bodies of dimension n>2 is proved. The first step is the reduction to the case when the convex body is o-symmetric and has axial rotational symmetry. This step works for related inequalities compatible with Steiner symmetrization. Secondly, for these convex bodies, a stability version of the characterization of ellipsoids by the fact that each hyperplane section is centrally symmetric is established.
We show a reverse isoperimetric inequality within the class of relative outer parallel bodies, with respect to a general convex body $E$, along with its equality condition. Based on the convexity of the sequence of quermassintegrals of Minkowski sums we also prove further inequalities.
We present a sharp version of the isoperimetric inequality for finitely generated groups due to T. Couhlon and L. Saloff-Coste based on the proof suggested by M. Gromov.
We prove a reverse isoperimetric inequality for domains homeomorphic to a disc with the boundary of curvature bounded below lying in two-dimensional Alexandrov spaces of curvature $\geqslant c$. We also study the equality case.
Let $G$ be a graph which satisfies $c^{-1} a^r \le |B(v,r)| \le c a^r$, for some constants $c,a>1$, every vertex $v$ and every radius $r$. We prove that this implies the isoperimetric inequality $|\partial A| \ge C |A| / \log(2+ |A|)$ for some constant $C=C(a,c)$ and every finite set of vertices $A$.
In this paper we study the quantitative isoperimetric inequality in the plane. We prove the existence of a set $Ω$, different from a ball, which minimizes the ratio $δ(Ω)/λ^2(Ω)$, where $δ$ is the isoperimetric deficit and $λ$ the Fraenkel asymmetry, giving a new proof ofthe quantitative isoperimetric inequality. Some new properties of the optimal set are also shown.
It is shown that every even, zonal measure on the Euclidean unit sphere gives rise to an isoperimetric inequality for sets of finite perimeter which directly implies the classical Euclidean isoperimetric inequality. The strongest member of this large family of inequalities is shown to be the only affine invariant one among them - the Petty projection inequality. As an application, a family of sharp Sobolev inequalities for functions of bounded variation is obtained, each of which is stronger than the classical Sobolev inequality. Moreover, corresponding families of Lp isoperimetric and Sobolev…
A Geometric proof for the Polygonal Isoperimetric Inequality
本文利用约束问题的最优性条件与三角形几何基本思想，从几何角度证明了在给定面积下最小化周长的多边形必定是正多边形。
We investigate isoperimetric constants of infinite tessellating metric graphs. We introduce a curvature-like quantity, which plays the role of a metric graph analogue of discrete curvature notions for combinatorial tessellating graphs. We then prove a lower estimate and a criterium for positivity of the isoperimetric constant.
A matroid polytope approach to sharp affine isoperimetric inequalities for volume decomposition functionals
本文通过拟阵多面体方法，证明体分解泛函的递推公式，建立其尖锐仿射等周不等式，并展示拟阵理论在凸几何中的应用。
This paper announces the discovery of an isoperimetric inequality for the area of plane regions defined by binary forms. This result has been applied subsequently in the enumeration of solutions to the Thue inequality and, given its fundamental nature, may find application in other areas as well.
We introduce tropical analogues of the notion of volume of polytopes, leading to a tropical version of the (discrete) classical isoperimetric inequality. The planar case is elementary, but a higher-dimensional generalization leads to an interesting class of ordinary convex polytopes, characterizing the equality case in the isoperimetric inequality. This study is motivated by open complexity questions concerning linear optimization and its tropical analogs.
We prove an isoperimetric inequality for the uniform measure on a uniformly convex body and for a class of uniformly log-concave measures (that we introduce). These inequalities imply (up to universal constants) the log-Sobolev inequalities proved by Bobkov--Ledoux as well as the isoperimetric inequalities due to Bakry-Ledoux and Bobkov--Zegarlinski. We also recover a concentration inequality for uniformly convex bodies, similar to that proved by Gromov--Milman.
We prove a discrete analogue to a classical isoperimetric theorem of Weil for surfaces with non-positive curvature. It is shown that hexagons in the triangular lattice have maximal volume among all sets of a given boundary in any triangulation with minimal degree 6.
Isoperimetric inequality for Finsler manifolds with non-negative Ricci curvature
本文证明了具有非负 Ricci 曲率和 Euclidean 体积增长的测度 Finsler 流形的尖锐等周不等式，并在附加假设下证明了其刚性结果，进而推导出 Euclidean 空间中锥的加权各向异性等周不等式在不可逆情形下的刚性。
We prove that if $(X,\mathsf d,\mathfrak m)$ is an essentially non-branching metric measure space with $\mathfrak m(X)=1$, having Ricci curvature bounded from below by $K$ and dimension bounded from above by $N \in (1,\infty)$, understood as a synthetic condition called Measure-Contraction property, then a sharp isoperimetric inequality à la Lévy-Gromov holds true. Measure theoretic rigidity is also obtained.
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…
We prove a sharp dimension-free isoperimetric inequality, involving the volume entropy, in non-compact metric measure spaces with non-negative synthetic Ricci curvature.
Isoperimetric inequality for non-Euclidean polygons
针对正多边形在固定周长与边数下面积最大这一经典等周结论，本文将其推广到非欧情形，并完整证明了双曲几何与球面几何中的多边形等周不等式。
We study harmonic and quasi-harmonic discs in metric spaces admitting a uniformly local quadratic isoperimetric inequality for curves. The class of such metric spaces includes compact Lipschitz manifolds, metric spaces with upper or lower curvature bounds in the sense of Alexandrov, some sub-Riemannian manifolds, and many more. In this setting, we prove local Hoelder continuity and continuity up to the boundary of harmonic and quasi-harmonic discs.
We prove a sharp isoperimetric inequality for the class of metric measure spaces verifying the synthetic Ricci curvature lower bounds $MCP(0,N)$ and having Euclidean volume growth at infinity. We avoid the classical use of the Brunn-Minkowski inequality, not available for $MCP(0,N)$, and of the PDE approach, not available in the singular setting. Our approach will be carried over by using a scaling limit of localisation.
A new proof of the Wulff-Gage isoperimetric inequality and its applications
本文通过给出原点对称凸体Wulff-Gage等周不等式的新证明，进而证明了log-Minkowski问题的唯一性，并为二维$C^{2}$边界凸体提供了曲率熵log-Minkowski不等式的新证明。
Our main result gives an improved bound on the filling areas of closed curves in Banach spaces which are not closed geodesics. As applications we show rigidity of Pu's classical systolic inequality and investigate the isoperimetric constants of normed spaces. The latter has further applications concerning the regularity of minimal surfaces in Finsler manifolds.
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 establish a quantitative isoperimetric inequality for weighted Riemannian manifolds with $\mathrm{Ric}_{\infty} \ge 1$. Precisely, we give an upper bound of the volume of the symmetric difference between a Borel set and a sub-level (or super-level) set of the associated guiding function (arising from the needle decomposition), in terms of the deficit in Bakry-Ledoux's Gaussian isoperimetric inequality. This is the first quantitative isoperimetric inequality on noncompact spaces besides Euclidean and Gaussian spaces. Our argument makes use of Klartag's needle decomposition (also called local…
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。
A strong quantitative form of the fractional isoperimetric inequality
本文提出分数阶定量等周不等式的强版本，推广了已有局部结果，采用与原方法核心思路不同的正则化流程完成证明，并据此推导得到分数阶Cheeger不等式的稳定性估计。
We obtain an explicit formula for comparing total curvature of level sets of functions on Riemannian manifolds, and develop some applications of this result to the isoperimetric problem in spaces of nonpositive curvature.
In this paper, we prove a class of weighted isoperimetric inequalities for bounded domains in hyperbolic space by using the isoperimetric inequality with log-convex density in Euclidean space. As a consequence, we remove the horo-convex assumption of domains in a weighted isoperimetric inequality proved by Scheuer-Xia. Furthermore, we prove weighted isoperimetric inequalities for star-shaped domains in warped product manifolds. Particularly, we obtain a weighted isoperimetric inequality for star-shaped hypersurfaces lying outside a certain radial coordinate slice in the anti-de Sitter-Schwarzs…
Affine variational functionals, isoperimetric problems and stability
本文研究三个仿射变分量的等周问题，证明椭球是唯一极值，并借助L_p Busemann--Petty质心不等式的稳定性获得定量等周不等式及仿射Kohler-Jobin不等式的稳定性结果。
In this paper we study the relationship of hyperbolicity and (Cheeger) isoperimetric inequality in the context of Riemannian manifolds and graphs. We characterize the hyperbolic manifolds and graphs (with bounded local geometry) verifying this isoperimetric inequality, in terms of their Gromov boundary. Furthermore, we characterize the trees with isoperimetric inequality (without any hypothesis). As an application of our results, we obtain the solvability of the Dirichlet problem at infinity for these Riemannian manifolds and graphs, and that the Martin boundary is homeomorphic to the Gromov b…
The normalised volume measure on the $\ell_p^n$ unit ball ($1\leq p\leq 2$) satisfies the following isoperimetric inequality: the boundary measure of a set of measure $a$ is at least $cn^{1/p}\tilde{a}\log^{1-1/p}(1/\tilde{a})$, where $\tilde{a}=\min(a,1-a)$.
We prove a generalized isoperimetric inequality for a domain diffeomorphic to a sphere that replaces filling volume with $k$-dilation. Suppose $U$ is an open set in $\mathbb{R}^n$ diffeomorphic to a Euclidean $n$-ball. We show that in dimensions at least 4 there is a map from a standard Euclidean ball of radius about $vol(\partial U)^{1/(n-1)}$ to $U$, with degree 1 on the boundary, and $(n-1)$-dilation bounded by some constant only depending on $n$. We also give an example in dimension 3 of an open set where no such map with small $(n-1)$-dilation can be found. The generalized isoperimetric i…
Affine Fractional Sobolev and Isoperimetric Inequalities
本文建立 $\mathbb R^n$ 上的尖锐仿射分数阶 Sobolev 不等式，强化并推出 Almgren–Lieb 不等式，极限下蕴含 Zhang 的仿射 Sobolev 不等式，同时获得分数阶 Petty 投影不等式并证明径向平均体猜想。
The Cartan-Hadamard isoperimetric inequality under controlled variation of curvature
本文在截面曲率受控变化条件下证明了任意维度的Cartan-Hadamard猜想，通过控制商极小化子的凸包并比较度量畸变与双曲等周轮廓盈余，给出关键证明步骤。
Optimal sets for the quantitative isoperimetric inequality in the plane with the barycentric distance
针对平面上关联等周亏量与重心距离的定量等周不等式，研究其最优集的存在性、正则性等性质，推导最优性条件并明确最优集的结构特征。
It is shown that if $C$ is an $n$-dimensional convex body then there is an affine image $\widetilde C$ of $C$ for which $${|\partial \widetilde C|\over |\widetilde C|^{n-1\over n}}$$ is no larger than the corresponding expression for a regular $n$-dimensional ``tetrahedron''. It is also shown that among $n$-dimensional subspaces of $L_p$ (for each $p\in [1,\infty]), \ell^n_p$ has maximal volume ratio.\vskip3in
Sharp and rigid isoperimetric inequality in metric measure spaces with non-negative Ricci curvature
利用最优传输理论，在Lott-Sturm-Villani意义下非负Ricci曲率的度量测度空间中，证明了涉及体积熵的尖锐无维数等周不等式，并刻画了等号成立条件。
This paper considers affine analogues of the isoperimetric inequality in the sense of piecewise linear topology. Given a closed polygon P embedded in R^d having n edges, we give upper and lower bounds for the minimal number of triangles needed to forma triangulated embedded orientable surface in R^d having P as its geometric boundary. The most interesting case is dimension dimension 3, where we give an upper bound of 7 n^2 triangles, and a lower bound for some polygons P that require at least 1/2 n^2 triangles. In dimension 2 and dimensions 5 and above one needs only O(n) triangles. The case o…
A reverse isoperimetric inequality for convex shapes with inclusion constraint
本文刻画了圆盘内固定面积凸形状的最大周长解，证明其为等边多边形，并给出边长的显式方程，同时探讨了一般包含形状的情形。
The isoperimetric inequality and CMC hypersurfaces in Cartan-Hadamard manifolds
本文通过常平均曲率超曲面的尖锐不等式及边界点对积分和Jacobi场估计，证明了$3\leq n\leq9$维Cartan-Hadamard流形中的欧几里得等周不等式，确立了相应维度的Cartan-Hadamard猜想。
Brunn-Minkowski and Reverse Isoperimetric Inequalities for Dual Quermassintegrals
本文在对偶Brunn-Minkowski理论中建立了两个新几何不等式：证明了Lutwak猜想的原点对称凸体对偶均质积分的Brunn-Minkowski不等式，并推广Ball体积比不等式得出在John位置下立方体最大化对偶均质积分的逆向等周不等式。
The Orlicz-Brunn-Minkowski theory receives considerable attention recently, and many results in the $L_p$-Brunn-Minkowski theory have been extended to their Orlicz counterparts. The aim of this paper is to develop Orlicz $L_ϕ$ affine and geominimal surface areas for single convex body as well as for multiple convex bodies, which generalize the $L_p$ (mixed) affine and geominimal surface areas -- fundamental concepts in the $L_p$-Brunn-Minkowski theory. Our extensions are different from the general affine surface areas by Ludwig (in Adv. Math. 224 (2010)). Moreover, our definitions for Orlicz $…
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 …
The $L_p$-floating area, curvature entropy, and isoperimetric inequalities on the sphere
该研究将欧氏几何中的$L_p$-仿射表面积等概念推广到球面空间，引入球面凸体的$L_p$-浮动面积与曲率熵，证明相关等周不等式、对偶公式与单调性，并推广到非负常曲率空间形式。
We prove an inequality for submanifolds of Cartan-Hadamard manifolds, which relates the geometry of a submanifold to the measure of the geodesics in the ambient space which it intersects. For hypersurfaces, this gives an extension of Banchoff and Pohl's isoperimetric inequality to spaces of non-positive curvature. We also prove a modified version of Croke's isoperimetric inequality for hypersurfaces immersed in Cartan-Hadamard manifolds and a sharp, quantitative version of an isoperimetric inequality of Yau in spaces of negative curvature. We discuss the relationship between these results, and…
Isoperimetric Inequalities in the $\ell_{\infty}$ cube and torus
本文证明了 $\ell_{\infty}$ 度量下立方体与环面的精确等周不等式，并作为推论恢复了 Bollobás-Leader 网格边等周不等式。
In this paper we prove an isoperimetric inequality of euclidean type for complete metric spaces admitting a cone-type inequality. These include all Banach spaces and all complete, simply-connected metric spaces of non-positive curvature in the sense of Alexandrov or, more generally, of Busemann. The main theorem generalizes results of Gromov and Ambrosio-Kirchheim.
In this paper we adapt the well-estabilished $Γ$-calculus techniques to the context of $RCD(K,\infty)$ spaces, proving Bobkov's local isoperimetric inequality and, when $K$ is positive, the Gaussian isoperimetric inequality in this class of spaces. The proof relies on the measure-valued $Γ_2$ operator introduced by Savaré.
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