A Brunn-Minkowski inequality for the integer lattice

A close discrete analog of the classical Brunn-Minkowksi inequality that holds for finite subsets of the integer lattice is obtained. This is applied to obtain strong new lower bounds for the cardinality of the sum of two finite sets, one of which has full dimension, and, in fact, a method for computing the exact lower bound in this situation, given the dimension of the lattice and the cardinalities of the two sets. These bounds in turn imply corresponding new bounds for the lattice point enumerator of the Minkowski sum of two convex lattice polytopes. A Rogers-Shephard type inequality for the lattice point enumerator in the plane is also proved.

     

Brunn-Minkowski inequalities for contingency tables and integer flows

We establish approximate log-concavity for a wide family of combinatorially defined integer-valued functions. Examples include the number of non-negative integer matrices (contingency tables) with prescribed row and column sums (margins), as a function of the margins and the number of integer feasible flows in a network, as a function of the excesses at the vertices. As a corollary, we obtain approximate log-concavity for the Kostant partition function of type A. We also present an indirect evidence that at least some of the considered functions might be genuinely log-concave.     

Some isoperimetric inequalities and eigenvalue estimates in weighted manifolds

In this paper we prove general inequalities involving the weighted mean curvature of compact submanifolds immersed in weighted manifolds. As a consequence we obtain a relative linear isoperimetric inequality for such submanifolds. We also prove an extrinsic upper bound to the first non-zero eigenvalue of the drift Laplacian on closed submanifolds of weighted manifolds.     

General Sobolev type inequalities for symmetric forms

A general Sobolev type inequality is introduced and studied for general symmetric forms by defining a new type of Cheeger's isoperimetric constant. Finally, concentration of measure for the Lp type logarithmic Sobolev inequality is presented.     

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.     

多重调和映射的等周型和Fejer-Riesz型不等式

本文讨论多重调和映射的等周型和Fejer-Riesz型不等式.首先,本文改进Kalaj和Meˇstrovi′c的相应结果,并将其结果推广到多重调和映射.其次,本文证明Pavlovi′c和Dostani′c的相应结果对于多重调和映射也是成立的.最后,本文建立关于多重调和映射的Fejer-Riesz型不等式.     

L_p-极投影Brunn-Minkowski不等式

将经典的对偶混合体积概念推广到L_p空间,提出了"q-全对偶混合体积"的概念.将传统的p≥1的L_p投影体概念拓展,提出p1时的L_p投影体和混合投影体概念,并且建立了L_p-极投影Brunn-Minkowski不等式.作为应用,推广了熟知的极投影Brunn-Minkowski不等式,获得了投影Brunn-Minkowski不等式的L_p空间的极形式.     

Isoperimetric type inequalities for harmonic functions

For 0<p<+∞ let hp be the harmonic Hardy space and let bp be the harmonic Bergman space of harmonic functions on the open unit disk U. Given 1?p<+∞, denote by ‖⋅bp_‖ and ‖⋅hp_‖ the norms in the spaces bp and hp, respectively. In this paper, we establish the harmonic hp-analogue of the known isoperimetric type inequality ‖fb2p_‖?‖fhp_‖, where f is an arbitrary holomorphic function in the classical Hardy space Hp. We prove that for arbitrary p>1, every function f∈hp satisfies the inequality

‖fb2p_‖?ap‖fhp_‖,     

对偶Brunn-Minkowski不等式的逆

20世纪80年代Milman曾指出:反向Brunn-Minkowski不等式是凸几何的一个深刻的结果.考虑了对偶情况,建立了一个反向的对偶Brunn-Minkowski不等式.进一步,均值积分差的反向对偶Brunn-Minkowski型不等式也被建立.     

Relative isoperimetric inequalities for minimal submanifolds outside a convex set

Let C be a closed convex set in a complete simply connected Riemannian manifold M with sectional curvature bounded above by a positive constant K. Assume that Σ is a compact minimal surface outside C such that Σ is orthogonal to ?C along ?Σ∩?C and ?Σ ～ ?C is radially connected from a point p ∈ ?Σ∩?C. We introduce a modified volume M_p(Σ) of Σ and obtain a sharp isoperimetric inequality where equality holds if and only if Σ is a geodesic half disk with constant Gaussian curvature K. We also prove higher dimensional isoperimetric inequalities for minimal submanifolds outside a closed convex set in a Riemannian manifold using the modified volume.     

Lp-极投影Brunn-Minkowski不等式

将经典的对偶混合体积概念推广到Lp空间,提出了"q-全对偶混合体积"的概念.将传统的P≥1的Lp投影体概念拓展,提出P<1时的Lp投影体和混合投影体概念,并且建立了Lp-极投影Brunn-Minkowski不等式.作为应用,推广了熟知的极投影Brunn-Minkowski不等式,获得了投影Brunn-Minkowski不等式的Lp空间的极形式.     

An isoperimetric comparison theorem for Schwarzschild space and other manifolds

We give a very general isoperimetric comparison theorem which, as an important special case, gives hypotheses under which the spherically symmetric -spheres of a spherically symmetric -manifold are isoperimetric hypersurfaces, meaning that they minimize -dimensional area among hypersurfaces enclosing the same -volume. This result greatly generalizes the result of Bray (Ph.D. thesis, 1997), which proved that the spherically symmetric 2-spheres of 3-dimensional Schwarzschild space (which is defined to be a totally geodesic, space-like slice of the usual -dimensional Schwarzschild metric) are isoperimetric. We also note that this Schwarzschild result has applications to the Penrose inequality in general relativity, as described by Bray.

     

On discrete and continuous wirtinger inequalities

We shall present several generalizations of discrete Wirtinger's inequality, and establish their continuous analogs.     

An isoperimetric inequality in the universal cover of the punctured plane

We find the largest ? (approximately 1.71579) for which any simple closed path α in the universal cover of R²?Z², equipped with the natural lifted metric from the Euclidean two-dimensional plane, satisfies L(α)≥?A(α), where L(α) is the length of α and A(α) is the area enclosed by α. This generalizes a result of Schnell and Segura Gomis, and provides an alternative proof for the same isoperimetric inequality in R²?Z².