Vertex Degrees of Steiner Minimal Trees in lpd and Other Smooth Minkowski Spaces

We find upper bounds for the degrees of vertices and Steiner points in Steiner Minimal Trees in the d-dimensional Banach spaces ldp 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 lp-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 (i.e. Minkowski 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.