Midpoint sets contained in the unit sphere of a normed space

TL;DR

This paper investigates the maximum size of point sets outside the unit ball in normed spaces whose midpoint sets lie entirely on the unit sphere, revealing geometric relationships in three dimensions and specific norms.

Contribution

It characterizes the maximum midpoint set size contained in the unit sphere in finite-dimensional normed spaces, highlighting differences between three and higher dimensions.

Findings

Maximum midpoint set size in 3D depends on the facial structure of the unit ball.

In higher dimensions, no direct relationship exists between the maximum size and the facial structure.

Exact maxima are determined for Euclidean and supremum norm spaces.

Abstract

The midpoint set M(S) of a set S of points is the set of all midpoints of pairs of points in S. We study the largest cardinality of a midpoint set M(S) in a finite-dimensional normed space, such that M(S) is contained in the unit sphere, and S is outside the closed unit ball. We show in three dimensions that this maximum (if it exists) is determined by the facial structure of the unit ball. In higher dimensions no such relationship exists. We also determine the maximum for euclidean and sup norm spaces.