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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Extremal problems in Minkowski space
related to minimal networks


Author: K. J. Swanepoel
Journal: Proc. Amer. Math. Soc. 124 (1996), 2513-2518
MSC (1991): Primary 52A40, 52A21, 49F10
MathSciNet review: 1327047
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Abstract: We solve the following problem of Z. Füredi, J. C. Lagarias and F. Morgan (1991): 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 they raised that arose out of the study of singularities in length-minimizing networks in Minkowski spaces.


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Additional Information

K. J. Swanepoel
Affiliation: Department of Mathematics and Applied Mathematics, University of Pretoria, Pretoria 0002, South Africa
Email: konrad@friedrichs.up.ac.za

DOI: http://dx.doi.org/10.1090/S0002-9939-96-03370-9
PII: S 0002-9939(96)03370-9
Keywords: Minimal networks, Minkowski spaces, finite-dimensional Banach spaces, sums of unit vectors problem
Received by editor(s): February 21, 1995
Communicated by: Peter Li
Article copyright: © Copyright 1996 American Mathematical Society