Sets of unit vectors with small pairwise sums

TL;DR

This paper investigates the maximum size of delta-additive sets of unit vectors in d-dimensional normed spaces, revealing exponential growth for delta>2/3, boundedness for delta<2/3, and linear growth at the critical threshold delta=2/3.

Contribution

It establishes the growth rates of delta-additive sets in normed spaces and identifies the precise maximum size at the critical threshold delta=2/3.

Findings

Maximum size grows exponentially for delta>2/3

Maximum size remains bounded for delta<2/3

Maximum size is exactly d at delta=2/3, except in l^1 space

Abstract

We study the sizes of delta-additive sets of unit vectors in a d-dimensional normed space: the sum of any two vectors has norm at most delta. One-additive sets originate in finding upper bounds of vertex degrees of Steiner Minimum Trees in finite dimensional smooth normed spaces (Z. F\"uredi, J. C. Lagarias, F. Morgan, 1991). We show that the maximum size of a delta-additive set over all normed spaces of dimension d grows exponentially in d for fixed delta>2/3, stays bounded for delta<2/3, and grows linearly at the threshold delta=2/3. Furthermore, the maximum size of a 2/3-additive set in d-dimensional normed space has the sharp upper bound of d, with the single exception of spaces isometric to three-dimensional l^1 space, where there exists a 2/3-additive set of four unit vectors.