Bounds on the roots of the Steiner polynomial

TL;DR

This paper establishes bounds on the roots of the Steiner polynomial for convex bodies in low dimensions, linking the roots' real parts to the body's principal radii of curvature.

Contribution

It provides new bounds on Steiner polynomial roots for convex bodies in dimensions up to five, connecting geometric curvature properties to polynomial root locations.

Findings

Real parts of roots are bounded by principal radii of curvature.

Bounds are valid for convex bodies in R^n with n ≤ 5.

The bounds relate geometric curvature to polynomial roots.

Abstract

We consider the Steiner polynomial of a C^2 convex body K in R^n (n \leq 5). The opposites of the real parts of the roots of the Steiner polynomial are bounded below by the minimum value and above by the maximum value of the principal radii of curvature of the boundary of K.