Chuanming Zong

Strange Phenomena in Convex and Discrete Geometry

1 Borsuks Problem.- x1 Introduction.- x2 The Perkal-Eggleston Theorem.- x3 Some Remarks.- x4 Larmans Problem.- x5 The Kahn-Kalai Phenomenon.- 2 Finite Packing Problems.- x1 Introduction.- x2 Supporting Functions, Area Functions, Minkowski Sums, Mixed Volumes, and Quermassintegrals.- x3 The Optimal Finite Packings Regarding Quermassintegrals.- x4 The L. Fejes Toth-Betke-Henk-Wills Phenomenon.- x5 Some Historical Remarks.- 3 The Venkov-McMullen Theorem and Steins Phenomenon.- x1 Introduction.- x2 Convex Bodies and Their Area Functions.- x3 The Venkov-McMullen Theorem.- x4 Steins Phenomenon.- x5 Some Remarks.- 4 Local Packing Phenomena.- x1 Introduction.- x2 A Phenomenon Concerning Blocking Numbers and Kissing Numbers.- x3 A Basic Approximation Result.- x4 Minkowskis Criteria for Packing Lattices and the Densest Packing Lattices.- x5 A Phenomenon Concerning Kissing Numbers and Packing Densities.- x6 Remarks and Open Problems.- 5 Category Phenomena.- x1 Introduction.- x2 Grubers Phenomenon.- x3 The Aleksandrov-Busemann-Feller Theorem.- x4 A Theorem of Zamfirescu.- x5 The Schneider-Zamfirescu Phenomenon.- x6 Some Remarks.- 6 The Busemann-Petty Problem.- x1 Introduction.- x2 Steiner Symmetrization.- x3 A Theorem of Busemann.- x4 The Larman-Rogers Phenomenon.- x5 Schneiders Phenomenon.- x6 Some Historical Remarks.- 7 Dvoretzkys Theorem.- x1 Introduction.- x2 Preliminaries.- x3 Technical Introduction.- x4 A Lemma of Dvoretzky and Rogers.- x5 An Estimate for ?V(AV).- x6 ?-nets and ?-spheres.- x7 A Proof of Dvoretzkys Theorem.- x8 An Upper Bound for M (n, ?).- x9 Some Historical Remarks.- Inedx.

What is known about unit cubes

Unit cubes, from any point of view, are among the simplest and the most important objects in n-dimensional Euclidean space. In fact, as one will see from this survey, they are not simple at all. On the one hand, the known results about them have been achieved by employing complicated machineries from Number Theory, Group Theory, Probability Theory, Matrix Theory, Hyperbolic Geometry, Combinatorics, etc.; on the other hand, the answers for many basic problems about them are still missing. In addition, the geometry of unit cubes does serve as a meeting point for several applied subjects such as Design Theory, Coding Theory, etc. The purpose of this article is to figure out what is known about the unit cubes and what do we want to know about them.

Sums, Projections, and Sections of Lattice Sets, and the Discrete Covariogram

AbstractBasic properties of finite subsets of the integer lattice ℤn are investigated from the point of view of geometric tomography. Results obtained concern the Minkowski addition of convex lattice sets and polyominoes, discrete X-rays and the discrete and continuous covariogram, the determination of symmetric convex lattice sets from the cardinality of their projections on hyperplanes, and a discrete version of Meyer’s inequality on sections of convex bodies by coordinate hyperplanes.

From deep holes to free planes

During the last decades, by applying techniques from Number Theory, Combinatorics and Measure Theory, remarkable progress has been made in the study of deep holes, free planes and related topics in packings of convex bodies, especially in lattice ball packings. Meanwhile, some fascinating new problems have been proposed. To stimulate further research in related areas, we will review the main results, some key techniques and some fundamental problems about deep holes, free cylinders and free planes in this paper. 0. Background and examples Question 0.1. In an equally populated region the government is going to open a large (fixed) number of schools. Of course, a pupil will be happy if he lives in the vicinity of a school. However, no matter how the schools are located, some of the pupils (in the deep holes) are farther away than the others. In this situation, how should the locations of the schools be chosen so as to minimize the schooling distance of those pupils? Let E denote the n-dimensional Euclidean space and let ‖x1,x2‖ denote the Euclidean distance between two points x1 and x2. For convenience, we call the minimum distance between distinct points of a set its separation. Based on Question 0.1 it is natural to ask the following question. Question 0.2. For which discrete planar set X of separation 2 does the function ρ(X) = sup y∈E2 min x∈X ‖y,x‖ attain its minimum? What is the minimum? By a routine argument one can show that the minimum is 2/ √ 3 and every optimal set is isometric to Λ2 = {z1(2, 0) + z2(1, √ 3) : zi ∈ Z}, where Z indicates the integer ring. In this case, one can triangulate E into congruent regular triangles of edge length 2 and with all their vertices belonging to Λ2. Received by the editors May 31, 2001, and, in revised form, January 1, 2002. 2000 Mathematics Subject Classification. Primary 05B40, 11H31, 52C15, 52C17. This work is supported by the National Science Foundation of China and a special grant from Peking University. c ©2002 American Mathematical Society

Some remarks concerning kissing numbers, blocking numbers and covering numbers

This article shows an inequality concerning blocking numbers and Hadwigers covering numbers and presents a strange phenomenon concerning kissing numbers and blocking numbers. As a simple corollary, we can improve the known upper bounds for Hadwigers covering numbers ford-dimensional centrally symmetric convex bodies to 3d−1.

The Blocking Numbers of Convex Bodies

Abstract. Besides determining the exact blocking numbers of cubes and balls, a conditional lower bound for the blocking numbers of convex bodies is achieved. In addition, several open problems are proposed.

The kissing numbers of tetrahedra

We determine the lattice kissing numbers of tetrahedra, by which we disprove a conjecture by Grünbaum. At the same time, we present a strange phenomenon concerning kissing numbers and packing densities of tetrahedra.

A Problem of Blocking Light Rays

Let K be an n-dimensional convex body with interior int(K). This article mainly deals with the following problem: How many nonoverlapping translates of K (or int(K)) are enough to block all the light rays (straight lines) starting from K?

Segments in ball packings

Denote by B n the n -dimensional unit ball centred at o. It is known that in every lattice packing of B n there is a cylindrical hole of infinite length whenever n ≥3. As a counterpart, this note mainly proves the following result: for any fixed e with e>0, there exist a periodic point set P ( n , e ) and a constant c ( n , e ) such that B n + P ( n , e ) is a packing in R n , and the length of the longest segment contained in R n \{int( eB n ) + P ( n , e )} is bounded by c ( n , e ) from above . Generalizations and applications are presented.

On the blocking number and the covering number of a convex body

In this paper, we introduce and study several numbers and functions associated to a convex body, by them we can study the covering number, the blocking number and even Borsuks partition number of a convex body in one setting.