对凸形内部一类特殊点的研究

1 简介对于一个凸形,其内部任一点都能表示为凸形的某条弦的中点,但是对于一般的凸形,什么样的点能表示为凸形的某个内接中心对称凸多边形的中心?本文将对这个命题的推广进行讨论.以下为本文的主要结论.定理 设Ω为平面上的凸形,定义T为Ω的所有内接中心对称凸多边形中心构成的集合,则图形T的面积S(T)满足0≤S(T)≤1/4S(Ω)不等式左端等号成立当且仅当Ω为中心对称图形,不等式右端等号成立当且仅当Ω为三角形.(以下如无特殊说明,“凸形”,“中心对称图形”均指平面上的图形,且不包括直线或直线的一部分.)     

平面Bonnesen型不等式

将用积分几何方法给出平面等周不等式以及Bonnesen型不等式,平面区域D的面积、周长、最大内接园半径及最小外接园半径的一些几何不等式的简单证明.     

N 元指数和对数平均的凸性及几何凸性

讨论了n 元指数平均和对数平均的凸性、S - 凸性、几何凸性及S - 几何凸性, 证明了:(1) n 元指数平均是S - 凹的和S - 几何凸的; (2) n 元第一对数平均是S - 凹的; (3) n 元第二对数平均是凹的和几何凸的. 最后提出了二个悬而未决的问题.     

关于曲率积分不等式的注记

本文主要研究平面卵形线的曲率积分不等式.利用积分几何中凸集的支持函数以及外平行集的性质,得到了Gage等周不等式与曲率的熵不等式的一个积分几何的简化证明;进一步地,我们得到了一个新的关于曲率积分的不等式.     

平面凸集存在最小凸生成集的充要条件

本文在平面上解决了StevenRLay在 [1 ]中提出的开放性问题“什么样的凸集存在唯一的最小凸生成子集” ,给出并证明了“平面上的凸集存在唯一的最小凸生成子集”的一个充要条件 .同时证明了En 中的开集一定不存在最小凸生成集 .     

Banach空间上凸集的凸性

以Banach空间的一般凸集为研究对象,将Banach空间的凸性研究推广到了内部非空的凸集上．打破了从单位球出发研究Banach空间几何的具有局限性的研究方法,给出了严格凸集的若干特征刻画及性质,并得到了严格凸集和光滑集之间的对偶定理．     

平面有限点集中的最大面积六边形

若平面上的有限点集构成凸多边形的顶点集,则称此有限点集处于凸位置令P表示平面上处于凸位置的有限点集,研究了P的子集所确定的凸六边形的面积与CH(P)面积比值的最大值问题.     

几何图形不等式漫谈(一)

<正>平面图形中的几何量,包含线段长度、角的大小及图形的面积.每类几何量之间除有相等关系之外,应该说多数情况下呈现的是不等关系.研究这些不等关系就构成了几何图形中不等式的内容(简称几何不等式).一种图形中的几何量若在某约束条件下它的值在一定范围变化,很自然地会提出什么时候这个量最大(或最小)的问题.这类问题与几何不     

常曲率平面上Fujiwara-Bol定理的注记

本文研究常曲率平面上的凸集,研究常曲平面上的凸集方法.根据Grinberg-Ren-Zhou的思想方法,我们给出著名的常曲率平面上Fujiwara-Bol定理的简单证明.     

积分几何不等式

本文研究n维欧氏空间中随机凸集同固定有界凸集相交的体积矩。在运用限弦投影的新方法和对称化原理对凸体内随机线段的运动测度和凸体内随机点偶的分布的极值性质做了深入讨论之后,建立了体积矩同凸集的体积之间的一系列积分几何不等式。经典的等周不等式同时获得新证明。     

On fencing problems

Fencing problems regard the bisection of a convex body in a way that some geometric measures are optimized. We introduce the notion of relative diameter and study bisections of centrally symmetric planar convex bodies, bisections by straight line cuts in general planar convex bodies and also bisections by hyperplane cuts for convex bodies in higher dimensions. In the planar case we obtain the best possible lower bound for the ratio between the relative diameter and the area.     

On heat conductors with a stationary hot spot

We consider a heat conductor having initial constant temperature and zero boundary temperature at every time.The hot spot is the point at which temperature attains its maximum at each given time. For convex conductors, if the hot spot does not move in time, we prove symmetry results for planar triangular and quadrangular conductors.Then, we examine the case of a general conductor and, by an asymptotic formula, we prove that, if there is a stationary critical point, not necessarily the hot spot, then the conductor must satisfy a geometric condition. In particular, we show that there is no stationary critical point inside planar non-convex quadrangular conductors. Mathematics Subject Classification (2000) Primary 35K05, 35K20, 35J05; Secondary 35J25, 35B38, 35B40     

Orlicz mixed affine quermassintegrals

A class of geometric quantities for convex bodies is introduced in the framework of Orlicz Brunn-Minkowski theory. It is shown that these new geometric quantities are affine invariant and precisely the generalizations of classical affine quermassintegrals.     

Checking the convexity of polytopes and the planarity of subdivisions

This paper considers the problem of verifying the correctness of geometric structures. In particular, we design simple optimal checkers for convex polytopes in two and higher dimensions, and for various types of planar subdivisions, such as triangulations, Delaunay triangulations, and convex subdivisions. Their performance is analyzed also in terms of the algorithmic degree, which characterizes the arithmetic precision required.     

How to Exhibit Toroidal Maps in Space

Steinitz's theorem states that a graph is the 1-skeleton of a convex polyhedron if and only if it is 3-connected and planar. The polyhedron is called a geometric realization of the embedded graph. Its faces are bounded by convex polygons whose points are coplanar. A map on the torus does not necessarily have such a geometric realization. In this paper we relax the condition that faces are the convex hull of coplanar points. We require instead that the convex hull of the points on a face can be projected onto a plane so that the boundary of the convex hull of the projected points is the image of the boundary of the face. We also require that the interiors of the convex hulls of different faces do not intersect. Call this an exhibition of the map. A map is polyhedral if the intersection of any two closed faces is simply connected. Our main result is that every polyhedral toroidal map can be exhibited. As a corollary, every toroidal triangulation has a geometric realization.     

On the number of disjoint convex quadrilaterals for a planar point set

For a given planar point set P, consider a partition of P into disjoint convex polygons. In this paper, we estimate the maximum number of convex quadrilaterals in all partitions.     

On a normed version of a Rogers–Shephard type problem

A translation body of a convex body is the convex hull of two of its translates intersecting each other. In the 1950s, Rogers and Shephard found the extremal values, over the family of n-dimensional convex bodies, of the maximal volume of the translation bodies of a given convex body. In our paper, we introduce a normed version of this problem, and for the planar case, determine the corresponding quantities for the four types of volumes regularly used in the literature: Busemann, Holmes–Thompson, and Gromov’s mass and mass*. We examine the problem also for higher dimensions, and for centrally symmetric convex bodies.     

Polynomially solvable special cases of the Steiner problem in planar networks

We give polynomial-time algorithms for two special cases of the Steiner problem: (1) the underlying network is planar and all terminals lie within a bounded number of "layers" of a single face, and (2) the problem is the rectilinear Steiner problem and the number of rectilinear convex hulls in the entire "onion" of convex hulls is bounded. Our algorithms build on well-known dynamic programming algorithms. For the second problem, we also use some geometric arguments.     

Hyperbolic metric,curvature of geodesics and hyperbolic discs in hyperbolic plane domains

The purpose of this paper is to investigate the relations among some geometric quantities defined for every hyperbolic plane domain of any connectivity, each of which measures, in some sense, how much the domain deviates either from a disc, convex domain, or simply connected domain on one hand, or a punctured domain on the other hand. Supported by the Landau Center for Mathematical Research in Analysis.     

一般平面曲线流的一点注记

讨论嵌入平面闭曲线的一般发展方程,并给出发展曲线的各种几何量的演化方程,然后证明发展方程的切向分量并不影响发展曲线的最终形状。