A Note on Borsuk’s Problem in Minkowski Spaces

Abstract

In 1993, Kahn and Kalai famously constructed a sequence of finite sets in d-dimensional Euclidean spaces that cannot be partitioned into less than \({{(1.203 \ldots + o(1))}^{{\sqrt d }}}\) parts of smaller diameter. Their method works not only for the Euclidean, but for all \({{\ell }_{p}}\)-spaces as well. In this short note, we observe that the larger the value of p, the stronger this construction becomes.