On Steiner entire function

TL;DR

This paper introduces the Steiner entire function, an analytic tool extending Steiner polynomials to infinite-dimensional convex sets, revealing geometric and probabilistic properties through complex analysis.

Contribution

It defines and analyzes the Steiner entire function, establishing fundamental properties, disproving a conjecture, and proposing new criteria for Gaussian continuity in infinite dimensions.

Findings

Characterizes Gaussian continuity via entire function growth

Disproves Gao and Vitale's conjecture

Proposes new criteria for intrinsic volume sequences

Abstract

We introduce and study the Steiner entire function, an analytic generating function for the intrinsic volumes of a convex compact set in a Hilbert space. This function extends the classical Steiner polynomial to infinite dimensions and encodes key geometric information about the set. We establish fundamental results on its order, type, and canonical product representation, and show how its analytic growth properties characterize Gaussian continuity. In particular, we provide new criteria for this property in terms of entire function theory, disprove a conjecture of Gao and Vitale, and conjecture a new characterization of admissible intrinsic volume sequences in infinite dimensions.