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Differentiability of quermassintegrals: A classification of convex bodies

Authors: M. A. Hernández Cifre and E. Saorín
Journal: Trans. Amer. Math. Soc. 366 (2014), 591-609
MSC (2010): Primary 52A20, 52A39; Secondary 52A40
Published electronically: July 24, 2013
MathSciNet review: 3130309
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Abstract: In this paper we characterize the convex bodies in $ \mathbb{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 $ \mathbb{R}^n$, for $ n\geq 3$, not satisfying the inradius condition in Teissier's problem on the geometric properties of the roots of the Steiner polynomial.

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Additional Information

M. A. Hernández Cifre
Affiliation: Departamento de Matemáticas, Universidad de Murcia, Campus de Espinardo, 30100-Murcia, Spain

E. Saorín
Affiliation: Institut für Algebra und Geometrie, Universität Magdeburg, Universitätsplatz 2, D-39106 Magdeburg, Germany

Keywords: Hadwiger classification, inner parallel body, Steiner polynomial, Teissier problem, inradius, quermassintegrals, tangential body, extreme vector, form body
Received by editor(s): February 8, 2010
Received by editor(s) in revised form: July 13, 2010, January 20, 2011, June 21, 2011, and October 17, 2011
Published electronically: July 24, 2013
Additional Notes: The authors were supported by Dirección General de Investigación (MICINN) MTM2009-10418 and by “Programa de Ayudas a Grupos de Excelencia de la Región de Murcia”, Fundación Séneca, Agencia de Ciencia y Tecnología de la Región de Murcia (Plan Regional de Ciencia y Tecnología 2007/2010), 04540/GERM/06
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.