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Curvatures of typical convex bodies-- the complete picture


Author: Rolf Schneider
Journal: Proc. Amer. Math. Soc. 143 (2015), 387-393
MSC (2010): Primary 52A20; Secondary 53A07
DOI: https://doi.org/10.1090/S0002-9939-2014-12245-3
Published electronically: September 12, 2014
MathSciNet review: 3272763
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Abstract: It is known that a typical $ n$-dimensional convex body, in the Baire category sense, has the property that its set of umbilics of zero curvature has full measure in the boundary of the body. We show that a typical convex body has in addition the following properties. The spherical image of the set of umbilics of zero curvature has measure zero. The set of umbilics of infinite curvature is dense in the boundary and uncountable and its spherical image has full measure in the unit sphere.


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Additional Information

Rolf Schneider
Affiliation: Mathematisches Institut, Albert-Ludwigs-Universität, Eckerstr. 1, D–79104, Freiburg i. Br., Germany

DOI: https://doi.org/10.1090/S0002-9939-2014-12245-3
Keywords: Typical convex body, Baire category, umbilic, generalized curvatures, zero curvature, infinite curvature, spherical image
Received by editor(s): April 1, 2013
Published electronically: September 12, 2014
Communicated by: Lei Ni
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.