Cardinalities of k-distance sets in Minkowski spaces

TL;DR

This paper investigates the maximum size of k-distance sets in Minkowski spaces, proving a conjecture for 2D and parallelotope unit balls, and providing bounds for general spaces.

Contribution

It confirms a conjecture about the maximum size of k-distance sets in certain Minkowski spaces and offers new bounds for more general cases.

Findings

Confirmed the conjecture in 2D spaces.

Confirmed the conjecture when the unit ball is a parallelotope.

Provided weaker upper bounds for general Minkowski spaces.

Abstract

A subset of a metric space is a k-distance set if there are exactly k non-zero distances occuring between points. We conjecture that a k-distance set in a d-dimensional Banach space (or Minkowski space), contains at most (k+1)^d points, with equality iff the unit ball is a parallelotope. We solve this conjecture in the affirmative for all 2-dimensional spaces and for spaces where the unit ball is a parallelotope. For general spaces we find various weaker upper bounds for k-distance sets.