Extremal Problems in Minkowski Space related to Minimal Networks

TL;DR

This paper establishes an upper bound of 2n on the size of certain sets of unit vectors in n-dimensional Minkowski spaces, characterizing when equality occurs and connecting to minimal network singularities.

Contribution

It proves a polynomial upper bound on the maximum size of sets with subset sum norms less than 1 in Minkowski spaces, characterizing the extremal case as the 0-infinity space.

Findings

Maximum set size is at most 2n.

Equality holds iff the space is 0-infinity.

Connections to singularities in length-minimizing networks.

Abstract

We solve the following problem of Z. F\"uredi, J. C. Lagarias and F. Morgan [FLM]: Is there an upper bound polynomial in $n$ for the largest cardinality of a set S of unit vectors in an n-dimensional Minkowski space (or Banach space) such that the sum of any subset has norm less than 1? We prove that |S|\leq 2n and that equality holds iff the space is linearly isometric to \ell^n_\infty, the space with an n-cube as unit ball. We also remark on similar questions raised in [FLM] that arose out of the study of singularities in length-minimizing networks in Minkowski spaces.