On the projections of convex lattice sets

As a discrete analogue of Aleksandrov＇s projection theorem, it is natural to ask the following question： Can an o-symmetric convex lattice set of the integer lattice Z n be uniquely determined by its lattice projection counts？ In 2005, Gardner, Gronchi and Zong discovered a counterexample with cardinality 11 in the plane. In this paper, we will show that it is the only counterexample in Z2, up to unimodular transformations and with cardinality not larger than 17.     

On a discrete version of alexandrov’s projection theorem

In this paper, we consider a discrete version of Aleksandrov＇s projection theorem. We prove that an origin-symmetric convex lattice set, whose lattice＇s y-coordinates＇ absolute values are not bigger than 2, can be uniquely determined by its lattice projection counts if its cardinality is not 11. This partly answers a question on the discrete version of Aleksandrov＇s projection theorem which was proposed by Gardner, Gronchi and Zong in 2005.     

A criterion for finite lattice coverings

For a centrally symmetric convex and a covering lattice L for K, a lattice polygon P is called a covering polygon, if . We prove that P is a covering polygon, if and only if its boundary bd(P) is covered by (L ∩ P) + K. Further we show that this characterization is false for non-symmetric planar convex bodies and in Euclidean d–space, d ≥ 3, even for the unit ball K = B ^d. This revised version was published online in June 2006 with corrections to the Cover Date.     

投影体与U(P)的极值性质

为了研究著名的Schneider投影问题,E.Lutwak,D.Yang和张高勇最近引进了一个关于对称多胞形的新的仿射不变量U(P),并提出了关于U(P)下界的猜想.本文就二维、三维在一特定条件下的情形给予了此猜想的肯定回答并给出了严格的数学证明.     

Inequalities for polars of mixed projection bodies

In 1993 Lutwak established some analogs of the Brunn-Minkowsi inequality and the Aleksandrov-Fenchel inequality for mixed projection bodies. In this paper, following Lutwak, we give their polars forms. Further, as applications of our methods, we give a generalization of Pythagorean inequality for mixed volumes.     

椭球的Orlicz投影体

本文研究了文献[1]所引入的Orlicz投影体问题.利用Orlicz投影体在线性变换下的不变性,获得了椭球的Orlicz投影体仍是椭球的结果.作为例子,计算了当取两个特定的凸函数时单位球的Orlicz投影体的支持函数.     

投影体的宽度积分和仿射表面积

本文运用凸几何分析理论,建立了投影体的宽度积分和仿射表面积的一些新型Brunn-Minkowski 不等式,这些结果改进了Lutwak的几个有用的定理.作为应用,进一步给出了混合投影体极的Brunn- Minkowski型不等式.     

Onp-quermassintegral differences function

In this paper we establish Minkowski inequality and Brunn-Minkowski inequality forp-quermassintegral differences of convex bodies. Further, we give Minkowski inequality and Brunn-Minkowski inequality for quermassintegral differences of mixed projection bodies.     

关于Petty投影不等式猜想Lp-版本的几个不等式

本文研究了关于投影不等式的Petty猜想这个凸体理论中的一个著名公开问题.利用凸体的Lp-Brunn-Minkowski-Firey理论,建立了Petty投影不等式猜想的Lp-版本的几个不同精度的不等式,推广了已有文献的结论.     

A Simplified Elementary Proof of Hadwiger's Volume Theorem

One of the most important results in geometric convexity is Hadwiger's characterization of quermassintergrals and intrinsic volumes. The importance lies in that Hadwiger's theorem provides straightforward proofs of numerous results in integral geometry such as the kinematic formulas [Santaló, L. A.: Integral Geometry and Geometric Probability, Addison-Wesley, 1976], the mean projection formulas for convex bodies [Schneider, R.: Convex Bodies: The Brunn—Minkowski Theory, Cambridge Univ. Press, 1993], and the characterization of totally invariant set functions of polynomial type [Chen, B. and Rota, G.-C.: Totally invariant set functions of polynomial type, Comm. Pure Appl. Math. 47 (1994), 187–197]. For a long time the only known proof of Hadwiger's theorem was his original one [Hadwiger, H.: Vorlesungen über Inhalt, Oberfläche and Isoperimetrie, Springer, Berlin, 1957] (long and not available in English), until a new proof was obtained by Klain [Klain, D. A.: A short proof of Hadwiger's characterization theorem, Mathematika 42 (1995), 329–339., Klain, D. A. and Rota, G.-C.: Introduction to Geometric Probability, Lezioni Lincee, Cambridge Univ. Press, 1997], using a result from spherical harmonics. The present paper provides a simplified and self-contained proof of Hadwiger's theorem.     

Reconstructing Boundary Points of Convex Sets from X-ray Pictures

A method for the reconstruction of boundary points of convex sets is provided starting from three X-ray pictures in two orthogonal directions and from a point.     

A short proof of the twelve-point theorem

We present a short elementary proof of the following twelve-point theorem. Let M be a convex polygon with vertices at lattice points, containing a single lattice point in its interior. Denote by m (respectively, m*) the number of lattice points in the boundary of M (respectively, in the boundary of the dual polygon). Then m + m* = 12.Translated from Matematicheskie Zametki, vol. 77, no. 1, 2005, pp. 117–120.Original Russian Text Copyright © 2005 by D. Repov, M. Skopenkov, M. Cencelj.This revised version was published online in April 2005 with a corrected issue number.     

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.     

Covering a Convex Polygon by Triangles

According to a theorem of A. V. Bogomolnaya, F. L. Nazarov and S. E. Rukshin, if n points are given inside a convex n-gon, then the points and the sides of the polygon can be numbered from 1 to n so that the triangles spanned by the ith point and the ith side(i=1....,n ) cover the polygon. In this paper, we prove that the same can be done without assuming that the given points are inside the convex n-gon. We also show that in the general case at least [(n/3)] mutually nonoverlapping triangles can be constructed in the same manner.     

Kubota公式的注记

本文研究了n维欧式空间中En中r-维平面的密度与(n-1)维球的体积元之间的关系.利用了复合叠加等的方法,得到了n维欧式空间En的凸体和任意n-r维投影空间Ln-r之间均值积分的关系,推广了Kubota公式得到的结果