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On the Local Convexity of Intersection Bodies of Revolution

Published online by Cambridge University Press:  20 November 2018

M. Angeles Alfonseca  and
Jaegil Kim
M. Angeles Alfonseca
Affiliation:
Department of Mathematics, North Dakota State University, Fargo ND 58018, USA. e-mail: maria.alfonseca@ndsu.edu
Jaegil Kim
Affiliation:
Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, AB. e-mail: jaegil@ualberta.ca
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Abstract

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One of the fundamental results in convex geometry is Busemann's theorem, which states that the intersection body of a symmetric convex body is convex. Thus, it is only natural to ask if there is a quantitative version of Busemann's theorem, i.e., if the intersection body operation actually improves convexity. In this paper we concentrate on the symmetric bodies of revolution to provide several results on the (strict) improvement of convexity under the intersection body operation. It is shown that the intersection body of a symmetric convex body of revolution has the same asymptotic behavior near the equator as the Euclidean ball. We apply this result to show that in sufficiently high dimension the double intersection body of a symmetric convex body of revolution is very close to an ellipsoid in the Banach–Mazur distance. We also prove results on the local convexity at the equator of intersection bodies in the class of star bodies of revolution.

Keywords

52A2052A3844A12convex bodiesintersection bodies of star bodiesBusemann's theoremlocal convexity
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 67 , Issue 1 , 01 February 2015 , pp. 3 - 27
Copyright
Copyright © Canadian Mathematical Society 2015

References

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