10-28
How to make quermassintegrals differentiable: solving a problem by Hadwiger
by M. A. Hernandez Cifre, E. Saorin Gomez
Preprint series: 10-28
- MSC:
- 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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