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

TL;DR

This paper investigates the properties of Steiner-Minkowski polynomials for convex sets in high-dimensional spaces, introduces a renormalization method, and explicitly computes limit entire functions for key convex families as dimension grows.

Contribution

It introduces a renormalization procedure for Steiner-Minkowski polynomials and derives explicit limit entire functions for several convex set families in high dimensions.

Findings

Renormalized Steiner-Minkowski polynomials form a normal family in the complex plane.

Explicit limit entire functions are obtained for Euclidean balls, cubes, cross-polytopes, and simplices.

The study extends understanding of convex set geometry in high-dimensional analysis.

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.