Extremal Problems in Minkowski Space related to Minimal Networks
Metric Geometry
2007-07-23 v1 Functional Analysis
Abstract
We solve the following problem of Z. F\"uredi, J. C. Lagarias and F. Morgan [FLM]: Is there an upper bound polynomial in 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.
Cite
@article{arxiv.0707.3052,
title = {Extremal Problems in Minkowski Space related to Minimal Networks},
author = {Konrad J Swanepoel},
journal= {arXiv preprint arXiv:0707.3052},
year = {2007}
}
Comments
6 pages. 11-year old paper. Implicit question in the last sentence has been answered in Discrete & Computational Geometry 21 (1999) 437-447