The reverse isoperimetric inequality for convex plane curves through a length-preserving flow

Archiv der Mathematik - By a length-preserving flow, we provide a new proof of a conjecture on the reverse isoperimetric inequality composed by Pan et al. (Math Inequal Appl 13:329–338,...     

John’s Ellipsoid and the Integral Ratio of a Log-Concave Function

We extend the notion of John’s ellipsoid to the setting of integrable log-concave functions. This will allow us to define the integral ratio of a log-concave function, which will extend the notion of volume ratio, and we will find the log-concave function maximizing the integral ratio. A reverse functional affine isoperimetric inequality will be given, written in terms of this integral ratio. This can be viewed as a stability version of the functional affine isoperimetric inequality.     

Stability of a reverse isoperimetric inequality

In this note we will present a stability property of the reverse isoperimetric inequality newly obtained in [S.L. Pan, H. Zhang, A reverse isoperimetric inequality for convex plane curves, Beiträge Algebra Geom. 48 (2007) 303-308], which states that if K is a convex domain in the plane with perimeter p(K) and area a(K), then one gets , where denotes the oriented area of the domain enclosed by the locus of curvature centers of the boundary curve ∂K, and the equality holds if and only if K is a circular disc.     

Isoperimetric inequalities and conjugate points on Lorentzian surfaces

Recently, an isoperimetric inequality for a sector on the Minkowski 2-spacetime has been derived by the method of parallels and the relativistic Gauss-Bonnet formula. In the present paper, we derive an isoperimetric inequality for a sector on a Lorentzian surface with curvatureK ≤ C. As a sector can be modeled by a geodesic variation of a timelike geodesic, our isoperimetric inequality gives an upper bound for the spacelike boundary of a sector. As an application of our results, we give an elementary proof of the existence of conjugate points on a Lorentzian surface with curvatureK ≤ C < 0 and we obtain an upper bound for the (timelike) diameter of a globally hyperbolic Lorentzian surface withK ≤ C < 0 by comparison of sectors.     

An isoperimetric inequality for uniformly log-concave measures and uniformly convex bodies

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 [S.G. Bobkov, M. Ledoux, From Brunn-Minkowski to Brascamp-Lieb and to logarithmic Sobolev inequalities, Geom. Funct. Anal. 10 (5) (2000) 1028-1052] and the isoperimetric inequalities due to Bakry, Ledoux [D. Bakry, M. Ledoux, Lévy-Gromov's isoperimetric inequality for an infinite-dimensional diffusion generator, Invent. Math. 123 (2) (1996) 259-281] and Bobkov, Zegarliński [S.G. Bobkov, B. Zegarliński, Entropy bounds and isoperimetry, Mem. Amer. Math. Soc. 176 (829) (2005), x+69]. We also recover a concentration inequality for uniformly convex bodies, similar to that proved by Gromov, Milman [M. Gromov, V.D. Milman, Generalization of the spherical isoperimetric inequality to uniformly convex Banach spaces, Compos. Math. 62 (3) (1987) 263-282].     

Ornstein-Uhlenbeck半群证明不等式的一些应用

按照Ornstein-Uhlenbeck的思想方法,用Ornstein-Uhlenbeck半群和Ornstein-Uhlenbeck算子的一些重要性质,对Brascamp-Lieb不等式、高斯对数Sobolev不等式、逆Bobkov等周不等式等几个重要的几何与分析不等式给出了另一证明.     

等周不等式与Bol-Fujiwara定理

本文研究著名的Bol-Fujiwara定理.利用积分几何方法和经典的等周不等式,得到了Bol-Fujiwara定理的一个推广(定理1),以及推广了的Bol-Fujiwara定理的逆定理(定理2).     

On the <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript> affine isoperimetric inequalities

We obtain an isoperimetric inequality which estimate the affine invariant p-surface area measure on convex bodies. We also establish the reverse version of L _p -Petty projection inequality and an affine isoperimetric inequality of Γ_− p K.     

Analytic inequalities,isoperimetric inequalities and logarithmic Sobolev inequalities

There is a simple equivalence between isoperimetric inequalities in Riemannian manifolds and certain analytic inequalities on the same manifold, more extensive than the familiar equivalence of the classical isoperimetric inequality in Euclidean space and the associated Sobolev inequality. By an isoperimetric inequality in this connection we mean any inequality involving the Riemannian volume and Riemannian surface measure of a subset α and its boundary, respectively. We exploit the equivalence to give log-Sobolev inequalities for Riemannian manifolds. Some applications to Schrödinger equations are also given.     

关于四面体的Bonnesen型等周不等式

主要研究几何体的Bonnesen型等周不等式.得到了两个关于四面体的Bonnesen型等周不等式;进一步地,给出了关于四面体的等周不等式的一个简单证明.     

Lévy-Gromov's isoperimetric inequality for an infinite dimensional diffusion generator

We establish, by simple semigroup arguments, a Lévy-Gromov isoperimetric inequality for the invariant measure of an infinite dimensional diffusion generator of positive curvature with isoperimetric model the Gaussian measure. This produces in particular a new proof of the Gaussian, isoperimetric inequality. This isoperimetric inequality strengthens the classical logarithmic Sobolev inequality in this context. A local version for the heat kernel measures is also proved, which may then be extended into an isoperimetric inequality for the Wiener measure on the paths of a Riemannian manifold with bounded Ricci curvature.Oblatum 19-VI-1995     

从积分几何的观点看几何不等式——纪念李国平院士吴新谋教授诞辰100周年

该文先介绍一些中国数学家在几何不等式方面的工作.作者用积分几何中著名的Poincarè公式及Blaschke公式估计一随机凸域包含另一域的包含测度, 得到了经典的等周不等式和Bonnesen -型不等式.还得到了一些诸如对称混合等周不等式、Minkowski -型和Bonnesen -型对称混合等似不等式在内的一些新的几何不等式.最后还研究了Gage -型等周不等式以及Ros -型等周不等式.     

R3中卯形闭曲面的等周亏格的上界的注记

利用R^3中卵形结果的高斯曲率不等式以及著名的等周不等式,将R^3中卵形闭曲面的高斯曲率K应用到空间曲面的等周亏格的上界估计中,得到了R^3中卵形闭曲面的等周亏格的一个新的上界,并给出其简单证明．     

Lévy–Gromov’s isoperimetric inequality for an infinite dimensional diffusion generator

 We establish, by simple semigroup arguments, a Lévy–Gromov isoperimetric inequality for the invariant measure of an infinite dimensional diffusion generator of positive curvature with isoperimetric model the Gaussian measure. This produces in particular a new proof of the Gaussian isoperimetric inequality. This isoperimetric inequality strengthens the classical logarithmic Sobolev inequality in this context. A local version for the heat kernel measures is also proved, which may then be extended into an isoperimetric inequality for the Wiener measure on the paths of a Riemannian manifold with bounded Ricci curvature. Oblatum 19-VI-1995     

Compactness theorems and an isoperimetric inequality for critical points of elliptic parametric functionals

We consider parametric variational double integrals with elliptic Lagrangians F depending on the surface normal and prove a compactness theorem for -critical immersions. As a key ingredient for the relevant a priori estimates we use F. Sauvigny's F-conformal parameters adapted to the parametric integrand F. As a by-product of our analysis we obtain an isoperimetric inequality for -critical immersions generalizing the classical isoperimetric inequality for minimal surfaces. Received November 19, 1999 / Accepted February 4, 2000 / Published online July 20, 2000     

Bonnesen-style inequalities and pseudo-perimeters for polygons

In this paper, we establish some Bonnesen-style isoperimetric inequalities for plane polygons via an analytic isoperimetric inequality and an isoperimetric inequality in pseudo-perimeters of polygons.1991 Mathematics Subject Classification 51M10, 51M25,52A40,26D10.     

A Sharp Stability Result for the Relative Isoperimetric Inequality Inside Convex Cones

The relative isoperimetric inequality inside an open, convex cone $\mathcal{C}$ states that, at fixed volume, $B_{r} \cap\mathcal{C}$ minimizes the perimeter inside $\mathcal{C}$ . Starting from the observation that this result can be recovered as a corollary of the anisotropic isoperimetric inequality, we exploit a variant of Gromov’s proof of the classical isoperimetric inequality to prove a sharp stability result for the relative isoperimetric inequality inside $\mathcal{C}$ . Our proof follows the line of reasoning in Figalli et al.: Invent. Math. 182:167–211 (2010), though several new ideas are needed in order to deal with the lack of translation invariance in our problem.     

From U-bounds to isoperimetry with applications to H-type groups

In this paper we study U-bounds in relation to L₁-type coercive inequalities and isoperimetric problems for a class of probability measures on a general metric space (RN,d). We prove the equivalence of an isoperimetric inequality with several other coercive inequalities in this general framework. The usefulness of our approach is illustrated by an application to the setting of H-type groups, and an extension to infinite dimensions.     

Petty's projection inequality and Santalo's affine isoperimetric inequality

Using results of K. Kiener and the Riesz-Sobolev convolution inequality we give a new proof of Petty's projection inequality. By the same method we also obtain a proof of Santalo's affine isoperimetric inequality.Supported in part by BSF and Erwin Schrödinger Auslandsstipendium J0630, J0804.     

Average linking with complex subspaces

We generalize Banchoff–Pohl?s isoperimetric inequality to complex affine space.