Pairwise intersecting homothets of a convex body

TL;DR

This paper establishes an upper bound on the number of pairwise intersecting homothets of a convex body in high-dimensional space and improves bounds for k-distance sets in Minkowski spaces.

Contribution

It provides a new upper bound for the maximum number of pairwise intersecting homothets of a convex body and enhances bounds for k-distance sets in Minkowski spaces.

Findings

Maximum of 3^{d+1} pairwise intersecting homothets in d-dimensional space.

Improved upper bounds for k-distance sets in Minkowski spaces.

Applicable to centrally symmetric convex bodies.

Abstract

We show that the maximum number of pairwise intersecting positive homothets of a $d$-dimensional centrally symmetric convex body, none of which contains the center of another in its interior, is at most $3^{d+1}$. Also, we improve upper bounds for cardinalities of $k$-distance sets in Minkowski spaces.