Results and Problems in Combinatorial Geometry

Abstract

In this short book, the authors discuss three types of problems from combinatorial geometry: Borsuk's partition problem, covering convex bodies by smaller homothetic bodies, and the illumination problem. They show how closely related these problems are to each other. The presentation is elementary, with no more than high-school mathematics and an interest in geometry required to follow the arguments. Most of the discussion is restricted to two- and three-dimensional Euclidean space, though sometimes more general results and problems are given. Thus even the mathematically unsophisticated reader can grasp some of the results of a branch of twentieth-century mathematics that has applications in such disciplines as mathematical programming, operations research and theoretical computer science. At the end of the book the authors have collected together a set of unsolved and partially solved problems that a sixth-form student should be able to understand and even attempt to solve.

Billiard representation for multidimensional cosmology with intersecting p-branes near the singularity

“…In [45] we proposed the simple geometric criterion for finiteness of volume of B and compactness ofB in terms of the positions of the points (4.30) with respect to the (N −2)- [47,48]. For the case of S N −2 this problem is equivalent to the problem of covering spheres with spheres [49,50].…”

A counterexample to Borsuk’s conjecture

“…The assertion of Borsuk's conjecture was proved in dimensions 2 and 3 and in all dimensions for centrally symmetric convex bodies and smooth convex bodies. See [9,1,4] and references cited there. Lassak [14] proved that f(d) < 2d~x + 1, and Schramm [16] showed that for every e , if d is sufficiently large, f(d) < (-^(3/2) + e)d .…”

Camera placement in integer lattices

“…In many cases (though not all) their analysis requires nothing more than basic topics from geometry, number theory, and graph theory and as such they are very well suited for a wide audience [2]. In recent years there has been particular emphasis on the algorithmic component of visibility problems in polygonal configurations and as such they have come to be studied under the area of "art gallery (watchman) problems."…”

“…In Section 3 we show that the cameras of an optimal configuration have to be visible modulo p for each prime p ~ s 11 d; in particular, a solution to the problem exists. Then we reformulate our problem into the following integer optimization problem: (2) where u' is an absolutely monotone function, B is a linear operator, and m E N; the three parameters u', B, and m depend on s (see Section 3 for the appropriate definitions of u', B, and m). This enables us to solve the problem, in Section 4, for the case of s::::; 3d cameras: a configuration of at most 3d cameras is optimal if and only if its cameras are evenly distributed in the classes of A/p as p ranges over the set of primes; in particular, a configuration of at most 2d cameras is optimal if and only if its cameras are pairwise visible.…”