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2010-28

How to make quermassintegrals differentiable: solving a problem by Hadwiger

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.