Vertex Degrees of Steiner Minimal Trees in ℓ p d and Other Smooth Minkowski Spaces
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We find upper bounds for the degrees of vertices and Steiner points in Steiner Minimal Trees (SMTs) 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 . Our upper bounds follow from characterizations of singularities of SMTs due to Lawlor and Morgan , 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 (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.