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

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

2008-03-04

arxiv.org/abs/0803.0443v1

Discrete & Computational Geometry 21 (1999) 437-447

Mathematics

Metric Geometry

12 pages

Scientific paper

10.1007/PL00009431

We 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.

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