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Intersection bodies and polytopes

Part of: General convexity

Published online by Cambridge University Press:  26 February 2010

Gaoyong Zhang
Gaoyong Zhang
Affiliation:
Department of Applied Mathematics and Physics, Polytechnic University, Brooklyn, NY 11201, U.S.A.gzhang@math.poly.edu
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Extract

An origin-symmetric convex body K in ℝn is called an intersection body if its radial function ρK is the spherical Radon transform of a non-negative measure µ on the unit sphere Sn−1. When µ is a positive continuous function, K is called the intersection body of a star body. The notion of intersection body was introduced by Lutwak [L]. It played a key role in the solution of the Busemann-Petty problem, see [G1], [G2], [L], [Z1] and [Z2]. Koldobsky [K] showed that the cross-polytope is an intersection body. This indicates that the statement in [Z3] that no origin-symmetric convex polytope in ℝn (n > 3) is an intersection body is not correct. This paper will prove the weaker statement that no origin-symmetric convex polytope in ℝn (n > 3) is the intersection body of a star body.

MSC classification

Secondary: 52A30: Variants of convex sets (star-shaped, ($m, n$)-convex, etc.)

Information

Type
Research Article
Information
Mathematika , Volume 46 , Issue 1 , June 1999 , pp. 29 - 34
Copyright
Copyright © University College London 1999

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