How to make quermassintegrals differentiable: solving a problem by Hadwiger

by M. A. Hernandez Cifre, E. Saorin Gomez


Preprint series: 10-28

52A20 Convex sets in $n$ dimensions (including convex hypersurfaces), See also {53A07, 53C45}
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.

Upload: 2010-07-28

Update: 2011 -01 -18


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