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

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References

CCampi, S.. Convex intersection bodies in three and four dimensions. Mathemaika (this volume), 15–27.CrossRefGoogle Scholar
FGW.Fallert, H., Goodey, P. and Weil, W.. Spherical projections and centrally symmetric sets. Adv. Math., 129 (1997), 301–322.CrossRefGoogle Scholar
G1.Gardner, R. J.. A positive answer to the Busemann-Petty problem in three dimensions. Annals Math., 140 (1994), 435–447.CrossRefGoogle Scholar
G2.Gardner, R. J.. Geometric Tomography. (Cambridge University Press, 1995).Google Scholar
H.Helgason, S.. Groups and Geometric Analysis. (Academic Press, 1984).Google Scholar
K.Lutkaw, E.. Intersection bodies, positive definite distributions and the Busemann-Petty problem. Amer. J. Math., 120 (1998), 827–840.Google Scholar
L.Lutkaw, E.. Intersection bodies and dual mixed volumes. Adv. Math., 71 (1988), 232–261.Google Scholar
S.Schneider, R.. Convex Bodies: The Brunn-Minkowski Theory. (Cambridge University Press, 1993).CrossRefGoogle Scholar
St.Strichartz, R.. LP estimates for Radon transforms in Euclidean and non-Euclidean spaces. Duke Math. J. 48 (1981), 699–727.CrossRefGoogle Scholar
Z1.Zhang, G.. Intersection bodies and the Busemann-Petty inequalities in ℝ4. Annals Math., 140 (1994), 331–346.CrossRefGoogle Scholar
Z2.Zhang, G.. A positive solution to the Busemann-Petty problem in ℝ4. Annals Math., 149 (1999), 535–543.CrossRefGoogle Scholar
Z3.Zhang, G.. Sections of convex bodies. Annals J Math. 118 (1996), 319–340.Google Scholar