Steiner polynomials via ultra-logconcave sequences

TL;DR

This paper studies the roots of relative Steiner polynomials of convex bodies, revealing their stability properties, geometric implications, and connection to ultra-logconcave sequences, with results extending to high dimensions.

Contribution

It characterizes the root structure of Steiner polynomials using ultra-logconcave sequences and establishes new stability and geometric properties across dimensions.

Findings

Roots cover the entire upper half-plane except the positive real axis as dimension increases.

Steiner polynomials are stable if and only if the dimension is at most 9.

Pairs of convex bodies with boundary roots satisfy specific Aleksandrov-Fenchel inequalities.

Abstract

We investigate structural properties of the cone of roots of relative Steiner polynomials of convex bodies. We prove that they are closed, monotonous with respect to the dimension, and that they cover the whole upper half-plane, except the positive real axis, when the dimension tends to infinity. In particular, it turns out that relative Steiner polynomials are stable polynomials if and only if the dimension is $\leq 9$. Moreover, pairs of convex bodies whose relative Steiner polynomial has a complex root on the boundary of such a cone have to satisfy some Aleksandrov-Fenchel inequality with equality. An essential tool for the proofs of the results is the characterization of Steiner polynomials via ultra-logconcave sequences.