Elsevier

Advances in Mathematics

Volume 226, Issue 6, 1 April 2011, Pages 5320-5337
Advances in Mathematics

The geometry of p-convex intersection bodies

Dedicated to the memory of Nigel J. Kalton, 1946–2010
https://doi.org/10.1016/j.aim.2011.01.011Get rights and content
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Abstract

Busemann's theorem states that the intersection body of an origin-symmetric convex body is also convex. In this paper we provide a version of Busemann's theorem for p-convex bodies. We show that the intersection body of a p-convex body is q-convex for certain q. Furthermore, we discuss the sharpness of the previous result by constructing an appropriate example. This example is also used to show that IK, the intersection body of K, can be much farther away from the Euclidean ball than K. Finally, we extend these theorems to some general measure spaces with log-concave and s-concave measures.

MSC

44A12
52A15
52A21

Keywords

p-Convex body
Star-shaped body
Intersection body
Busemann's theorem

Cited by (0)

The first and third authors were supported in part by U.S. National Science Foundation grant DMS-0652684. The second author was supported in part by NSERC.