This is an announcement for the paper "Vertex degrees of Steiner minimal trees in $\ell_p^d$ and other smooth Minkowski spaces" by K. J. Swanepoel. 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. Archive classification: math.MG math.FA Mathematics Subject Classification: 05C05 (Primary); 49Q10 (Secondary) Citation: Discrete & Computational Geometry 21 (1999) 437-447 Remarks: 12 pages The source file(s), steiner-lp.tex: 30143 bytes, is(are) stored in gzipped form as 0803.0443.gz with size 10kb. The corresponding postcript file has gzipped size 81kb. Submitted from: konrad.swanepoel@gmail.com The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/0803.0443 or http://arXiv.org/abs/0803.0443 or by email in unzipped form by transmitting an empty message with subject line uget 0803.0443 or in gzipped form by using subject line get 0803.0443 to: math@arXiv.org.