关于凸体覆盖的Hadwiger猜想的两个等价形式

令K是一个内部(记作intK)包含原点o的凸体,bdK为其边界,m为覆盖K所需的intK的平移的最小个数.本文证明,存在正实数η和含于η(bdK)的m元点集C1使得C1+int K覆盖K;存在正实数η′、实数γ∈(0,1)和含于η′(bd K)的m元点集C2使得C2+γK覆盖K.基于这两个事实,本文得到关于凸体覆盖的Hadwiger猜想的两个等价形式.本文还引入一个可以替代宗传明提出的攻克Hadwiger猜想的数量方案中的γm(K)的新泛函.     

Rienmannian Submersions and Isoperimetric Inequalities

Isoperimetric constants of the total spaces of Riemannian submersions are estimated in terms of those of the basis and fibers.     

A Discrete Isoperimetric Problem

We prove that the perimeter of any convex n-gons of diameter 1 is at most n2nsin (/2n). Equality is attained here if and only if n has an odd factor. In the latter case, there are (up to congruence) only finitely many extremal n-gons. In fact, the convex n-gons of diameter 1 and perimeter n2n sin (/2n) are in bijective correspondence with the solutions of a diophantine problem.     

Isoperimetric Inequalities on Compact Manifolds

We prove an isoperimetric inequality on compact Riemannian manifolds corresponding to the limit case of a scale of optimal Sobolev inequalities.     

Reflection Theorems and the Tame Kernel of a Number Field

Let L be a number field containing the pth primitive root of unity ζ _p . We investigate the p-rank of the ideal class groups of some subfields of L by using reflection theorems and establish relations between the p-rank of the ideal class groups and that of groups of units of some subfields of L.

Let F be a number field and 𝒪 _F the ring of integers in F. We also study the p-rank of tame kernels of F and establish relations between the p-rank of K ₂𝒪 _F and that of some direct summands of the ideal class group of F(ζ _p ).     

The carvingwidth of hypercubes

The notion of the carvingwidth of a graph was introduced by Seymour and Thomas [Call routing and the ratcatcher, Combinatorica 14 (1994) 217-241]. In this note, we show that the carvingwidth of a d-dimensional hypercube equals 2^d-1.     

经典等周不等式的又一简单初等的证明

In this short note we will give another simple and elementary proof of the classical isoperimetric inequality in the Euclidean plane.     

The Smallest Convex Cover for Triangles of Perimeter Two

We find the unique smallest convex region in the plane that contains a congruent copy of every triangle of perimeter two. It is the triangle ABC with AB=2/3, B=60°, and BC1.00285.     

Rudolph diagrams and an analytic realization of the Vitushkin covering

An example of a three-sheeted covering over the ball in ² is constructed. The covering is the union of a four-dimensional ball with an analytic disk. The projection of the ball is a local homeomorphism ramified along the disk, whose projection is a part of an algebraic curve.Translated fromMatematicheskie Zametki, Vol. 60, No. 2, pp. 206–224, August, 1996.This research was partially supported by the Russian Foundation for Basic Research under grant No. 93-011-225.     

Nonlinear complementarity problem and systems of convex inequalities

LetH be a nonempty closed convex subset of a topological vector spaceE andF be a real-valued function onH × H Then, we prove that, under some conditions, there existsx ₀H such thatF(x ₀,y)0 for ally H. Furthermore, we obtain a necessary and sufficient condition that a finite system of convex inequalities is irreducibly inconsistent.This work was supported in part by the Matsunaga Research Grant. The author wishes to express his sincere thanks to Professor H. Umegaki for his invaluable suggestions and advice.     

An Isoperimetric Inequality for Hyperplane Sections of a Convex Body

We obtain an inequality which estimates the product of the volumes of the two parts into which a convex body is divided by a hyperplane through its interior in terms of the volume of the hyperplane section. The basic tool in the proof is a Localization lemma for log-concave functions due to Kannan, Lovász and Simonovits.     

On the motion of rigid bodies in a viscous incompressible fluid

This paper is a survey on classical results and open questions about minimization problems concerning the lower eigenvalues of the Laplace operator. After recalling classical isoperimetric inequalities for the two first eigenvalues, we present recent advances on this topic. In particular, we study the minimization of the second eigenvalue among plane convex domains. We also discuss the minimization of the third eigenvalue. We prove existence of a minimizer. For others eigenvalues, we just give some conjectures. We also consider the case of Neumann, Robin and Stekloff boundary conditions together with various functions of the eigenvalues.AMS Subject Classification: 49Q10m, 35P15, 49J20.     

The optimal constant in Wente's $ L^\infty $ estimate

We explore some geometric aspects of compensation compactness associated to Jacobian determinants. We provide the optimal constant in Wente's inequality - the original motivation of this work - and go on to give various extensions to geometric situations. In fact we improve Wente's inequality somewhat, making it more appropriate for applications in which optimal results are required. This is demonstrated when we prove an optimal inequality for immersed surfaces of constant mean curvature in , contolling their diameter in terms of their area and curvature. Received: November 4, 1996