A note on Borsuk's problem in Minkowski spaces

TL;DR

This paper discusses how a construction related to Borsuk's problem becomes more effective in Minkowski spaces as the parameter p increases, extending previous Euclidean space results.

Contribution

It shows that the known partitioning difficulty in Euclidean spaces extends and intensifies in $ extit{ ext{l}}_p$-spaces with larger p values, highlighting a new aspect of Borsuk's problem.

Findings

Construction's strength increases with p in $ extit{ ext{l}}_p$-spaces.

Extension of Euclidean results to Minkowski spaces.

Implication for Borsuk's problem in different normed 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.