A counterexample to Borsuk's conjecture

Abstract

Let $f(d)$ be the smallest number so that every set in $R^d$ of diameter 1 can be partitioned into $f(d)$ sets of diameter smaller than 1. Borsuk's conjecture was that $f(d)=d+1$. We prove that $f(d)\ge (1.2{)}^{\sqrt{d}}$ for large~$d$.

3 pages