On the location of roots of Steiner polynomials

Abstract

We investigate the roots of relative Steiner polynomials of convex bodies. In dimension 3 we give a precise description of their location in the complex plane and we study the analogous problem in higher dimensions. In particular, we show that the roots (in the upper half plane) form a convex cone; for dimensions ≤ 9 this cone is completely contained in the (open) left half plane, which is not true in dimensions ≥ 12. Moreover, we characterize certain special families of convex bodies by means of properties of their roots.