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

TL;DR

This paper establishes upper bounds on the degrees of vertices and Steiner points in Steiner Minimal Trees within smooth Minkowski spaces, showing these bounds are independent of dimension, unlike in minimal spanning trees.

Contribution

It extends characterizations of SMT singularities and derives new dimension-independent upper bounds for vertex degrees in smooth Banach spaces.

Findings

Upper bound of d+1 for vertex degrees in smooth d-dimensional Banach spaces

Degree bounds are independent of dimension, contrasting with minimal spanning trees

Derived bounds using -summing norms and space inequalities

Abstract

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.