The local Steiner problem in finite-dimensional normed spaces

Details The local Steiner problem in finite-dimensional normed spaces The local Steiner problem in finite-dimensional normed spaces

2006-03-27

arxiv.org/abs/math/0603616v2

Discrete Comput Geom 37:419--442 (2007)

Mathematics

Metric Geometry

22 pages, 3 figures. Corollary 8 of the previous version is the new Theorem 7

Scientific paper

10.1007/s00454-006-1298-z

We develop a general method of proving that certain star configurations in finit e-dimensional normed spaces are Steiner minimal trees. This method generalizes the results of Lawlor and Morgan (1994) that could only be applied to differentiable norms. The generalization uses the subdifferential calculus from convex analysis. We apply this method to two special norms. The first norm, occurring in the work of Cieslik, has unit ball the polar of the difference body of the n-simplex (in dimension 3 this is the rhombic dodecahedron). We determine the maximum degree of a given point in a Steiner minimal tree in this norm. The proof makes essential use of extremal finite set theory. The second norm, occuring in the work of Mark Conger (1989), is the sum of the ell_1-norm and a small multiple of the ell_2 norm. For the second norm we determine the maximum degree of a Steiner point.

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