Steiner-Minkowski Polynomials of Convex Sets in High Dimension, and Limit Entire Functions

2007/09/01 by Victor Katsnelson, Katsnelson, Victor
Mathematics · #30-xx (Primary) #51 (Secondary) #Complex Variables (math.CV) #FOS: Mathematics #Metric Geometry (math.MG) #math.CV #math.MG #msc:30-xx #msc:51

paper · pdf · doi:10.48550/arxiv.0709.0037

15 pages

arxiv created 2007/09/01 · arxiv updated 2009/12/01

Abstract

For a convex set (K) of the (n)-dimensional Euclidean space, the Steiner-Minkowski polynomial (M_K(t)) is defined as the (n)-dimensional Euclidean volume of the neighborhood of the radius (t). Being defined for positive (t), the Steiner-Minkowski polynomials are considered for all complex (t). The renormalization procedure for Steiner polynomial is proposed. The renormalized Steiner-Minkowski polynomials corresponding to all possible solid convex sets in all dimensions form a normal family in the whole complex plane. For each of the four families of convex sets: the Euclidean balls, the cubes, the regular cross-polytopes and the regular symplexes of dimensions (n), the limiting entire functions, as (n) tends to infinity, are calculated explicitly.

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