The local Steiner problem in finite-dimensional normed spaces

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dc.creatorSwanepoel, Konrad J
dc.date2006-03-27
dc.date2006-07-19
dc.date.accessioned2026-07-07T07:51:40Z
dc.date.available2026-07-07T07:51:40Z
dc.descriptionWe 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.
dc.description22 pages, 3 figures. Corollary 8 of the previous version is the new Theorem 7
dc.identifierhttps://arxiv.org/abs/math/0603616
dc.identifierhttp://arxiv.org/abs/math/0603616
dc.identifierDiscrete Comput Geom 37:419--442 (2007)
dc.identifierdoi:10.1007/s00454-006-1298-z
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/125597
dc.subjectMetric Geometry
dc.subjectCombinatorics
dc.subject49Q10 (Primary); 05C05, 05D05, 52A21, 52A41, 52B40 (Secondary)
dc.titleThe local Steiner problem in finite-dimensional normed spaces
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