Vertex degrees of Steiner Minimal Trees in $\ell_p^d$ and other smooth Minkowski spaces

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dc.creatorSwanepoel, K. J.
dc.date2008-03-04
dc.date.accessioned2026-07-07T09:24:34Z
dc.date.available2026-07-07T09:24:34Z
dc.descriptionWe find upper bounds for the degrees of vertices and Steiner points in Steiner Minimal Trees in the d-dimensional Banach spaces \ell_p^d independent of d. This is in contrast to Minimal Spanning Trees, where the maximum degree of vertices grows exponentially in d (Robins and Salowe, 1995). Our upper bounds follow from characterizations of singularities of SMT's due to Lawlor and Morgan (1994), which we extend, and certain \ell_p-inequalities. We derive a general upper bound of d+1 for the degree of vertices of an SMT in an arbitrary smooth d-dimensional Banach space; the same upper bound for Steiner points having been found by Lawlor and Morgan. We obtain a second upper bound for the degrees of vertices in terms of 1-summing norms.
dc.description12 pages
dc.identifierhttps://arxiv.org/abs/0803.0443
dc.identifierhttp://arxiv.org/abs/0803.0443
dc.identifierDiscrete & Computational Geometry 21 (1999) 437-447
dc.identifierdoi:10.1007/PL00009431
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/156142
dc.subjectMetric Geometry
dc.subjectFunctional Analysis
dc.subject05C05 (Primary); 49Q10 (Secondary)
dc.titleVertex degrees of Steiner Minimal Trees in $\ell_p^d$ and other smooth Minkowski spaces
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