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 ${\mathbb{R}}_{\ge 0}$, 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 ${\mathbb{R}}_{\le 0}$, 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 $\{m_i:i=0,1,\dots \}$ to be the moments of a certain measure on ${\mathbb{R}}_{\ge 0}$, a question regarding the so called (Stieltjes) moment problem.