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4 - Some old and new problems in combinatorial geometry I: around Borsuk's problem

Published online by Cambridge University Press:  05 July 2015

By
Gil Kalai
Edited by
Artur Czumaj ,
Agelos Georgakopoulos ,
Daniel Král ,
Vadim Lozin  and
Oleg Pikhurko
Artur Czumaj
Affiliation:
University of Warwick
Agelos Georgakopoulos
Affiliation:
University of Warwick
Daniel Král
Affiliation:
University of Warwick
Vadim Lozin
Affiliation:
University of Warwick
Oleg Pikhurko
Affiliation:
University of Warwick
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Summary

Abstract

Borsuk [16] asked in 1933 if every set of diameter 1 in Rd can be covered by d + 1 sets of smaller diameter. In 1993, a negative solution, based on a theorem by Frankl and Wilson [42], was given by Kahn and Kalai [65]. In this paper I will present questions related to Borsuk's problem.

1 Introduction

The title of this paper is borrowed from Paul Erdős who used it (or a similar title) in many lectures and papers, e.g., [39]. I will describe several open problems in the interface between combinatorics and geometry, mainly convex geometry. In this part, I describe and pose questions related to the Borsuk conjecture. The selection of problems is based on my own idiosyncratic tastes. For a fuller picture, the reader is advised to read the review papers on Borsuk's problem and related questions by Raigorodskii [105, 106, 107, 110, 112]. Among other excellent sources are [13, 14, 18, 89, 99].

Karol Borsuk [16] asked in 1933 if every set of diameter 1 in Rd can be covered by d + 1 sets of smaller diameter. That the answer is positive was widely believed and referred to as the Borsuk conjecture. However, some people, including Rogers, Danzer, and Erdős, suggested that a counterexample might be obtained from some clever combinatorial configuration. In hindsight, the problem is related to several questions that Erdős asked and its solution was a great triumph for Erdősian mathematics.

2 Better lower bounds to Borsuk's problem

The asymptotics

Let f(d) be the smallest integer such that every set of diameter one in Rd can be covered by f(d) sets of smaller diameter. The set of vertices of a regular simplex of diameter one demonstrates that f(d) ≥ d + 1. The famous Borsuk–Ulam theorem [16] asserts that the d-dimensional ball of diameter 1 cannot be covered by d sets of smaller diameter.

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Surveys in Combinatorics 2015 , pp. 147 - 174
Publisher: Cambridge University Press
Print publication year: 2015

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