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 μμ on the non-negative real line R≥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 R≤0, and strictly increasing in the dimension, for any measure μμ. Moreover, it is proved that the 'smallest' cone of roots of these μμ-polynomials is the one given by the Steiner polynomial, which provides, for example, additional information about the roots of μμ-polynomials when the dimension is large enough. It will also give necessary geometric conditions for a sequence {mi:i=0,1,…} to be the moments of a certain measure on R≥0, a question regarding the so called (Stieltjes) moment problem.