Revista Matemática Iberoamericana

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Volume 31, Issue 2, 2015, pp. 477–496
DOI: 10.4171/RMI/842

Published online: 2015-07-16

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

María A. Hernández Cifre[1] and Jesús Yepes Nicolás[2]

(1) Universidad de Murcia, Spain
(2) Universidad Autónoma de Madrid, Spain

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.

Keywords: Generalized Wills functional, location of roots, weighted Steiner polynomial, measures, Stieltjes moment problem

Hernández Cifre María, Yepes Nicolás Jesús: On the roots of generalized Wills $\mu$-polynomials. Rev. Mat. Iberoam. 31 (2015), 477-496. doi: 10.4171/RMI/842