On the location of roots of Steiner polynomials (Q2430813)

The authors investigate roots of the Steiner polynomials of convex bodies in \(n\)-dimensional Euclidean space relative to a fixed convex gauge body \(E\). They show that the set \(R(n,E)\) of all such roots in the upper complex half plane is a convex cone, and determine \(R(3,E)\) explicitly. It is known (see, e.g.~\textit{B. Teissier} [Semin. differential geometry, Ann. Math. Stud. 102, 85--105 (1982; Zbl 0494.52009)]) that \(R(n,E)\) is contained in the upper left quadrant of the complex plane when \(n\leq 5\). In an earlier paper, the authors [Rev. Mat. Iberoam. 24, 631--644 (2008; Zbl 1158.52003)] showed this to be false for \(n\geq 12\). The present paper narrows this gap showing that \(R(n,E)\) is contained in the upper (strictly) left quadrant for \(n\leq9\). Finally certain classes of convex bodies are characterized by means of properties of their roots.