# On the roots of generalized Wills $μ$-polynomials

### María A. Hernández Cifre

Universidad de Murcia, Spain### Jesús Yepes Nicolás

Universidad Autónoma de Madrid, Spain

## Abstract

We investigate the roots of a family of geometric polynomials of convex bodies associated to a given measure $μ$ 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 $μ$. Moreover, it is proved that the 'smallest' cone of roots of these $μ$-polynomials is the one given by the Steiner polynomial, which provides, for example, additional information about the roots of $μ$-polynomials when the dimension is large enough. It will also give necessary geometric conditions for a sequence ${m_{i}:i=0,1,…}$ to be the moments of a certain measure on $R_{≥0}$, a question regarding the so called (Stieltjes) moment problem.

## Cite this article

María A. Hernández Cifre, Jesús Yepes Nicolás, On the roots of generalized Wills $μ$-polynomials. Rev. Mat. Iberoam. 31 (2015), no. 2, pp. 477–496

DOI 10.4171/RMI/842