Curvatures of typical convex bodies— the complete picture
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- by Rolf Schneider
- Proc. Amer. Math. Soc. 143 (2015), 387-393
- DOI: https://doi.org/10.1090/S0002-9939-2014-12245-3
- Published electronically: September 12, 2014
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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.References
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Bibliographic Information
- Rolf Schneider
- Affiliation: Mathematisches Institut, Albert-Ludwigs-Universität, Eckerstr. 1, D–79104, Freiburg i. Br., Germany
- MR Author ID: 199426
- ORCID: 0000-0003-0039-3417
- Received by editor(s): April 1, 2013
- Published electronically: September 12, 2014
- Communicated by: Lei Ni
- © Copyright 2014
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - 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
- MathSciNet review: 3272763