JournalsrmiVol. 31, No. 2pp. 477–496

On the roots of generalized Wills μ\mu-polynomials

  • María A. Hernández Cifre

    Universidad de Murcia, Spain
  • Jesús Yepes Nicolás

    Universidad Autónoma de Madrid, Spain
On the roots of generalized Wills $\mu$-polynomials cover
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Abstract

We investigate the roots of a family of geometric polynomials of convex bodies associated to a given measure μ\mu on the non-negative real line R0\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 R0\mathbb R_{\leq0}, and strictly increasing in the dimension, for any measure μ\mu. Moreover, it is proved that the 'smallest' cone of roots of these μ\mu-polynomials is the one given by the Steiner polynomial, which provides, for example, additional information about the roots of μ\mu-polynomials when the dimension is large enough. It will also give necessary geometric conditions for a sequence {mi ⁣:i=0,1,}\{m_i\colon i=0,1,\dots\} to be the moments of a certain measure on R0\mathbb R_{\geq0}, 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\mu-polynomials. Rev. Mat. Iberoam. 31 (2015), no. 2, pp. 477–496

DOI 10.4171/RMI/842