Extremal Problems in Minkowski Space related to Minimal Networks

Details Extremal Problems in Minkowski Space related to Minimal Networks Extremal Problems in Minkowski Space related to Minimal Networks

2007-07-20

arxiv.org/abs/0707.3052v1

Proceedings of the American Mathematical Society 124 (1996) 2513-2518

Mathematics

Metric Geometry

6 pages. 11-year old paper. Implicit question in the last sentence has been answered in Discrete & Computational Geometry 21 (

Scientific paper

10.1090/S0002-9939-96-03370-9

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.

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