We investigate structural properties of the cone of roots of relative Steiner polynomials of convex bodies. We prove that they are closed, monotonous with respect to the dimension, and that they cover the whole upper half-plane, except the positive real axis, when the dimension tends to infinity. In particular, it turns out that relative Steiner polynomials are stable polynomials if and only if the dimension is ≤ 9. Moreover, pairs of convex bodies whose relative Steiner polynomial has a complex root on the boundary of such a cone have to satisfy some Aleksandrov-Fenchel inequality with equality. An essential tool for the proofs of the results is the characterization of Steiner polynomials via ultra-logconcave sequences.This expression is called Minkowski-Steiner formula or relative Steiner formula of K. The coefficients W i (K; E) are the relative quermassintegrals of K, and they are a special case of the more general defined mixed volumes for which we refer to [18, s. 5.1]. In particular, we have W 0 (K; E) = vol(K),2000 Mathematics Subject Classification. Primary 52A20, 52A39; Secondary 30C15.

Abstract. In this paper we study the problem of classifying the convex bodies in R n , depending on the differentiability of their associated quermassintegrals with respect to the one-parameter-depending family given by the inner and outer parallel bodies. This problem was originally posed by Hadwiger in the 3-dimensional space. We characterize one of the non-trivial classes and give necessary conditions for a convex body to belong to the others. We also consider particular families of convex bodies, e.g. polytopes and tangential bodies.

In this paper we study the geometric meaning of the roots of the Steiner polynomial in the 3-dimensional space. We give a complete characterization of the convex bodies in R 3 depending on the type of roots of their Steiner polynomials. Furthermore, we show that these roots are also related to the famous Blaschke problem and the Teissier conjecture.

Abstract. Due to the "uncontrollable behavior" of the inner parallel bodies of a convex body K ⊂ R n regarding its boundary structure, it is not possible to get precise formulae for their volume/quermassintegrals, contrary to the case of the outer parallel bodies. In this paper we provide (sharp) bounds for the quermassintegrals of the inner parallel bodies, studying previously some particular properties of their boundary in terms of their outer normal vectors.

Abstract. In this paper we characterize the convex bodies in R n whose quermassintegrals satisfy certain differentiability properties, which answers a question posed by Bol in 1943 for the 3-dimensional space. This result will have unexpected consequences on the behavior of the roots of the Steiner polynomial: we prove that there exist many convex bodies in R n , for n ≥ 3, not satisfying the inradius condition in Teissier's problem on the geometric properties of the roots of the Steiner polynomial.