Remote Access Proceedings of the American Mathematical Society
Green Open Access

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
DOI: https://doi.org/10.1090/S0002-9939-96-03370-9
MathSciNet review: 1327047
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.


References [Enhancements On Off] (What's this?)

  • [BG] M. W. Bern and R. L. Graham, The shortest-network problem, Scientific American (January 1989), 66--71.
  • [BZ] Yu. D. Burago and V. A. Zalgaller, Geometric inequalities, Springer-Verlag, Berlin, Heidelberg, New York, 1988. MR 89b:52020
  • [C] J. A. Clarkson, Uniformly convex spaces, Trans.Amer. Math. Soc. 40 (1936), 396--414.
  • [CR] R. Courant and H. Robbins, What is Mathematics?, Oxford Univ. Press, Oxford, 1941. MR 3:144
  • [FLM] Z. Füredi, J. C. Lagarias and F. Morgan, Singularities of minimal surfaces and networks and related extremal problems in Minkowski space, Discrete and Computational Geometry (New Brunswick, NJ, 1989/1990), DIMACS Ser. Discrete Math. Theoret. Comput. Sci. 6, (J. E. Goodman, R. Pollack and W. Steiger, eds.), Amer. Math. Soc., Providence, RI, 1991, pp. 95--109. MR 93d:52009
  • [H] O. Hanner, On the uniform convexity of $L^{p}$ and $\ell ^{p}$, Ark. Mat. 3 (1956), 239--244. MR 17:987
  • [LM] G. Lawlor and F. Morgan, Paired calibrations applied to soap films, immiscible fluids, and surfaces and networks minimizing other norms, Pacific J. Math. 166 (1994), 55--82. MR 95i:58051
  • [M] F. Morgan, Minimal surfaces, crystals, networks, and undergraduate research, Math. Intelligencer 14 (1992), 37--44. MR 93h:53012
  • [Pe] C. M. Petty, Equilateral sets in Minkowski spaces, Proc. Amer. Math. Soc. 29 (1971), 369--374. MR 43:1051
  • [Pi] A. Pietsch, Operator Ideals, V. E. B. Deutscher Verlag Wiss., Berlin, 1978, and North-Holland, Amsterdam, 1980. MR 81a:47002; MR 81j:47001
  • [vLW] J. H. van Lint and R. M. Wilson, A Course in Combinatorics, Cambridge Univ. Press, Cambridge, 1992. MR 94g:05003

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 52A40, 52A21, 49F10

Retrieve articles in all journals with MSC (1991): 52A40, 52A21, 49F10


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: https://doi.org/10.1090/S0002-9939-96-03370-9
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

American Mathematical Society