The Borsuk Problem Conquered

Abstract

As we discussed in Section 14 of Chapter 4, Karel Borsuk formulated his celebrated conjecture in 1933: Borsuk’s Conjecture 22.1. For any bounded figure F in R ⁿ, a(F) ≤ N + 1, i.e., F can be decomposed into n + 1 parts of smaller diameters. For decades, everyone thought the conjecture was true but no one was able to prove it. Since you likely did not pay too much attention to the forewords, let me quote here from Paul Erdős’s foreword to the first edition of this book.