Borsuk's Problem in Metric Spaces

TL;DR

This paper extends Borsuk's problem to general metric spaces, proving that any bounded set can be partitioned into a specific number of smaller diameter subsets, generalizing previous conjectures.

Contribution

The paper proves a new upper bound for dividing bounded sets into smaller diameter subsets in n-dimensional metric spaces, generalizing Borsuk's problem.

Findings

Bounded sets in metric spaces can be divided into a finite number of smaller diameter subsets.

The number of subsets needed is bounded by a function involving 2^n and logarithmic factors.

The result generalizes classical Borsuk's problem from Euclidean spaces to metric spaces.

Abstract

In 1933, K. Borsuk proposed the following problem: Can every bounded set in $\mathbb{E}^n$ be divided into $n+1$ subsets of smaller diameters? In 1965, V. G. Boltyanski and I. T. Gohberg made the following conjecture: Every bounded set in an $n$-dimensional metric space can be divided into $2^n$ subsets of smaller diameters. In this paper, we prove the following result: Every bounded set in an $n$-dimensional metric space can be divided into $2^{n}((n+1)\log (n+1)+(n+1)\log \log (n+1)+5n+5)$ subsets of smaller diameters.