Notes on the roots of Steiner polynomials

We study the location and the size of the roots of Steiner polynomials ofconvex bodies in the Minkowski relative geometry. Based on a problem ofTeissier on the intersection numbers of Cartier divisors of compact algebraicvarieties it was conjectured that these roots have certain geometric propertiesrelated to the in- and circumradius of the convex body. We show that the rootsof 1-tangential bodies fulfill the conjecture, but we also present convexbodies violating each of the conjectured properties.