On universal covers for four-dimensional sets of a given diameter

TL;DR

This paper investigates conditions under which certain four-dimensional centrally symmetric polytopes serve as universal covers for all sets of unit diameter, establishing new bounds based on the number of facets.

Contribution

It proves that all centrally symmetric four-dimensional polytopes with at most fourteen facets are universal covers for unit diameter sets, complementing previous results for polytopes with more facets.

Findings

Polytopes with ≤14 facets are universal covers

Polytopes with >20 facets are not universal covers

Results specify facet count bounds for universality

Abstract

Makeev proved that among centrally symmetric four-dimensional polytopes, with more than twenty facets and circumscribed about the Euclidean ball of diameter one, there is no universal cover for the family of unit diameter sets. In this paper we examine the converse problem, and prove that each centrally symmetric polytope, with at most fourteen facets and circumscribed about the Euclidean ball of diameter one, is a universal cover for the family of unit diameter sets.