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

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.

💡 Research Summary

The paper studies the Steiner‑Minkowski polynomial (M_{K}(t)) associated with a convex body (K\subset\mathbb R^{n}). By definition, (M_{K}(t)) is the (n)-dimensional Lebesgue measure of the Minkowski sum (K+tB^{n}), where (B^{n}) denotes the unit Euclidean ball and (t\ge0). The classical Steiner formula tells us that this volume is a polynomial of degree (n): \