On the roots of generalized Wills $\mu$-polynomials

Abstract

We investigate the roots of a family of geometric polynomials of convex bodies associated to a given measure \mu on the non-negative real line \mathbb R_{\geq 0} , which arise from the so called Wills functional. We study its structure, showing that the set of roots in the upper half-plane is a closed convex cone, containing the non-positive real axis \mathbb R_{\leq0} , and strictly increasing in the dimension, for any measure \mu . Moreover, it is proved that the 'smallest' cone of roots of these \mu -polynomials is the one given by the Steiner polynomial, which provides, for example, additional information about the roots of \mu -polynomials when the dimension is large enough. It will also give necessary geometric conditions for a sequence \{m_i\colon i=0,1,\dots\} to be the moments of a certain measure on \mathbb R_{\geq0} , a question regarding the so called (Stieltjes) moment problem.