by Hernández Cifre, M. A., Saorín Gómez, E..

**Series:** 2010-28, Preprints

- MSC:
- 52A20 Convex sets in $n$ dimensions (including convex hypersurfaces)
- 52A39 Mixed volumes and related topics
- 52A40 Inequalities and extremum problems

**Abstract:**

In this paper we characterize the convex bodies in R^n whose

quermassintegrals satisfy certain differentiability properties, which fully

solves a problem posed by Hadwiger in R^3. 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 any n>=3, not satisfying Teissier's problem on the geometric properties of the roots of the Steiner polynomial related to the inradius of the set.

**Keywords:**

Hadwiger problem, inner parallel body, Steiner polynomial, Teissier problem, inradius, quermassintegrals, tangential body, extreme vector, form body.