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 Free Access

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 for the largest cardinality of a set of unit vectors in an -dimensional Minkowski space (or Banach space) such that the sum of any subset has norm less than 1? We prove that and that equality holds iff the space is linearly isometric to , the space with an -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.

**[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*, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 285, Springer-Verlag, Berlin, 1988. Translated from the Russian by A. B. Sosinskiĭ; Springer Series in Soviet Mathematics. MR**936419****[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., vol. 6, Amer. Math. Soc., Providence, RI, 1991, pp. 95–109. MR**1143291****[H]**O. Hanner,*On the uniform convexity of and ,*Ark. Mat.**3**(1956), 239--244. MR**17:987****[LM]**Gary Lawlor and Frank Morgan,*Paired calibrations applied to soap films, immiscible fluids, and surfaces or networks minimizing other norms*, Pacific J. Math.**166**(1994), no. 1, 55–83. MR**1306034****[M]**Frank Morgan,*Minimal surfaces, crystals, shortest networks, and undergraduate research*, Math. Intelligencer**14**(1992), no. 3, 37–44. MR**1184317**, https://doi.org/10.1007/BF03025868**[Pe]**C. M. Petty,*Equilateral sets in Minkowski spaces*, Proc. Amer. Math. Soc.**29**(1971), 369–374. MR**0275294**, https://doi.org/10.1090/S0002-9939-1971-0275294-8**[Pi]**Albrecht Pietsch,*Operator ideals*, Mathematische Monographien [Mathematical Monographs], vol. 16, VEB Deutscher Verlag der Wissenschaften, Berlin, 1978. MR**519680**

Albrecht Pietsch,*Operator ideals*, North-Holland Mathematical Library, vol. 20, North-Holland Publishing Co., Amsterdam-New York, 1980. Translated from German by the author. MR**582655****[vLW]**J. H. van Lint and R. M. Wilson,*A course in combinatorics*, Cambridge University Press, Cambridge, 1992. MR**1207813**

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