On H.Weyl and J.Steiner polynomials

TL;DR

This paper explores the roots of Steiner and Weyl polynomials, revealing their geometric origins, explicit calculations in examples, and the nature of their roots, including conditions for roots to be real or imaginary.

Contribution

It provides explicit formulas for Steiner and Weyl polynomials in natural examples and analyzes the location and nature of their roots, highlighting differences and relations between the two classes.

Findings

Steiner polynomial roots can have negative real parts.

Weyl polynomial roots are purely imaginary and simple.

Root patterns vary depending on geometric configurations.

Abstract

The paper deals with root problems for two classes of univariate polynomials both of geometric origin. The first class discussed, the class of Steiner polynomial, consists of polynomials, each associated with a compact convex set V in R^n. A polynomial of this class describes the volume of the set V+tB^n as a function of t, where t is a positive number and B^n denotes the unit ball in R. The second class, the class of Weyl polynomials, consists of polynomials, each associated with a Riemannian manifold M, where M} is isometrically embedded with positive codimension in R^n. A Weyl polynomial describes the volume of a tubular neighborhood of its associated M as a function of the tube's radius. These polynomials are calculated explicitly in a number of natural examples such as balls, cubes, squeezed cylinders. Furthermore, we examine how the above mentioned polynomials are related to one another and how they depend on the standard embedding of R^n into R^m for m>n. We find that in some cases the real part of any Steiner polynomial root will be negative. In certain other cases, a Steiner polynomial will have only real negative roots. In all of this cases, it can be shown that all of a Weyl polynomial's roots are simple and, furthermore, that they lie on the imaginary axis. At the same time, in certain cases the above pattern does not hold.