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.     

A new isoperimetric inequality for product measure and the tails of sums of independent random variables

In previous work we showed how very precise information on the tails of sums of (possibly Banach-space valued) random variables can be deduced from isoperimetric inequalities for product measure. We present here a new isoperimetric inequality, with a very simple proof, that allows the recovery of these bounds.Work partially supported by an NSF grant     

Extremal properties of some geometric processes

In this paper some isoperimetric inequalities for stationary random tessellations are discussed. At first, classical results on deterministic tessellations in the Euclidean plane are extended to the case of random tessellations. An isoperimetric inequality for the random Poisson polygon is derived as a consequence of a theorem of Davidson concerning an extremal property of tessellations generated by random lines inR ². We mention extremal properties of stationary hyperplane tessellations inR ^d related to Davidson's result in cased=2. Finally, similar problems for random arrangements ofr-flats inR ^d are considered (r).This work was done while the author was visiting the University of Strathclyde in Glasgow.     

The Dirichlet problem at infinity for random walks on graphs with a strong isoperimetric inequality

Summary We study the spatial behaviour of random walks on infinite graphs which are not necessarily invariant under some transitive group action and whose transition probabilities may have infinite range. We assume that the underlying graphG satisfies a strong isoperimetric inequality and that the transition operatorP is strongly reversible, uniformly irreducible and satisfies a uniform first moment condition. We prove that under these hypotheses the random walk converges almost surely to a random end ofG and that the Dirichlet problem forP-harmonic functions is solvable with respect to the end compactification If in addition the graph as a metric space is hyperbolic in the sense of Gromov, then the same conclusions also hold for the hyperbolic compactification in the place of the end compactification. The main tool is the exponential decay of the transition probabilities implied by the strong isoperimetric inequality. Finally, it is shown how the same technique can be applied to Brownian motion to obtain analogous results for Riemannian manifolds satisfying Cheeger's isoperimetric inequality. In particular, in this general context new (and simpler) proofs of well known results on the Dirichlet problem for negatively curved manifolds are obtained.The first author was partially supported by Consiglio Nazionale delle Ricerche, GNAFA Current address: Department of Mathematics, University of Edinburgh, Edinburgh EH9 3JZ, Scotland     

Concentration of norms and eigenvalues of random matrices

We prove concentration results for ?_pⁿ operator norms of rectangular random matrices and eigenvalues of self-adjoint random matrices. The random matrices we consider have bounded entries which are independent, up to a possible self-adjointness constraint. Our results are based on an isoperimetric inequality for product spaces due to Talagrand.     

Asymptotic Shapes of large Cells in Random Tessellations

We establish a close relationship between isoperimetric inequalities for convex bodies and asymptotic shapes of large random polytopes, which arise as cells in certain random mosaics in d-dimensional Euclidean space. These mosaics are generated by Poisson hyperplane processes satisfying a few natural assumptions (not necessarily stationarity or isotropy). The size of large cells is measured by a class of general functionals. The main result implies that the asymptotic shapes of large cells are completely determined by the extremal bodies of an inequality of isoperimetric type, which connects the size functional and the expected number of hyperplanes of the generating process hitting a given convex body. We obtain exponential estimates for the conditional probability of large deviations of zero cells from asymptotic or limit shapes, under the condition that the cells have large size. This work was supported in part by the European Network PHD, FP6 Marie Curie Actions, RTN, Contract MCRN-511953. Received: May 2005 Accepted: September 2005     

Wittmann's Law of Iterated Logarithm for Tail Sums of B-Valued Random Variables

We present an analogue of Wittmann's law of iterated logarithm (LIL) for tail sums of independent B-valued random variables by using the isoperimetric method and give the precise value of the upper limit for the LIL for tail sums.     

Anchored expansion and random walk

This paper studies anchored expansion, a non-uniform version of the strong isoperimetric inequality. We show that every graph with i-anchored expansion contains a subgraph with isoperimetric (Cheeger) constant at least i. We prove a conjecture by Benjamini, Lyons and Schramm (1999) that in such graphs the random walk escapes with a positive lim inf speed. We also show that anchored expansion implies a heat-kernel decay bound of order exp(—cn ^1/3). Submitted: September 1999, Revision: January 2000.     

Logarithmic Sobolev,isoperimetry and transport inequalities on graphs

In this paper,we study some functional inequalities(such as Poincaré inequality,logarithmic Sobolev inequality,generalized Cheeger isoperimetric inequality,transportation-information inequality and transportation-entropy inequality) for reversible nearest-neighbor Markov processes on connected finite graphs by means of(random) path method.We provide estimates of the involved constants.     

On exactness and isoperimetric profiles of discrete groups

We introduce a quasi-isometry invariant related to Property A and explore its connections to various other invariants of finitely generated groups. This allows to establish a direct relation between asymptotic dimension on one hand and isoperimetry and random walks on the other. We apply these results to reprove sharp estimates on isoperimetric profiles of some groups and to answer some questions in dimension theory.     

Log-concave and spherical models in isoperimetry

We derive several functional forms of isoperimetric inequalities, in the case of concave isoperimetric profile. In particular, we answer the question of a canonical and sharp functional form of the Lévy—Schmidt theorem on spheres. We use these results to derive a comparison theorem for product measures: the isoperimetric function of is bounded from below in terms of the isoperimetric functions of . We apply this to measures with finite dimensional isoperimetric behaviors. All the previous estimates can be improved when uniform enlargement is considered. Submitted: December 2000, Revised: May 2001.     

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     

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     

An elementary analysis of a procedure for sampling points in a convex body

In this paper we describe a new method for proving the polynomial-time convergence of an algorithm for sampling (almost) uniformly at random from a convex body in high dimension. Previous approaches have been based on estimating conductance via isoperimetric inequalities. We show that a more elementary coupling argument can be used to give a similar result. © 1998 John Wiley & Sons, Inc. Random Struct. Alg., 12, 213–235, 1998     

A new affine invariant for polytopes and Schneider's projection problem

New affine invariant functionals for convex polytopes are introduced. Some sharp affine isoperimetric inequalities are established for the new functionals. These new inequalities lead to fairly strong volume estimates for projection bodies. Two of the new affine isoperimetric inequalities are extensions of Ball's reverse isoperimetric inequalities.

     

Isoperimetry of group actions

The isoperimetric profile of a discrete group was introduced by Vershik, however it is well defined only for a restrictive class amenable groups. We generalize the notion of isoperimetric profile beyond the world of amenable groups by defining isoperimetric profiles of amenable actions of finitely generated groups on compact topological spaces. This allows to extend the definition of the isoperimetric profile to all groups which are exact in such a way that for amenable groups it is equal to Vershik's isoperimetric profile. The main feature of our construction is that it preserves many of the properties known from the classical case. We use these results to compute exact asymptotics of the isoperimetric profiles for several classes of non-amenable groups.     

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.     

两平面凸域的对称混合等周不等式

设K_k(k=i,j)为欧氏平面R~2中面积为A_k,周长为P_k的域,它们的对称混合等周亏格(symmetric mixed isoperimetric deficit)为σ(K_i,K_j)=P_i~2P_j~2-16π~2A_iA_j.根据周家足,任德麟(2010)和Zhou,Yue(2009)中的思想,用积分几何方法,得到了两平面凸域的Bonnesen型对称混合不等式及对称混合等周不等式,给出了两域的对称混合等周亏格的一个上界估计.还得到了两平面凸域的离散Bonnesen型对称混合不等式及两凸域的对称混合等周亏格的一个上界估计,并应用这些对称混合(等周)不等式估计第二类完全椭圆积分.     

Isoperimetric inequality in the Grushin plane

We prove a sharp isoperimetric inequality in the Grushin plane and compute the corresponding isoperimetric sets.