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Annals of Mathematics, II. Series Vol. 149, No. 2, pp. 691703 (1999) 

An analytic solution to the BusemannPetty problem on sections of convex bodiesR.J. Gardner, A. Koldobsky and T. SchlumprechtReview from Zentralblatt MATH: Let $K$ be an originsymmetric star body in $\Bbb R^n$ with $C^\infty$ boundary, and let $k\in\Bbb N \cup\{0\}, k\neq n  1.$ Suppose that $\xi\in S^{n1},$ and let $A_\xi$ be the corresponding parallel section function of $K.$ The function $A_\xi$ (or $(n  1)$dimensional $X$ray) gives the ($(n  1)$dimensional) volumes of all hyperplane sections of the body orthogonal to a given direction. The authors derive a formula connecting the derivatives of $A_\xi$ with the Fourier transform (in the sense of distributions) of powers $(\rho_K^{nk1})^\wedge$ of the radial function $\rho_K$ of the body: $$(\rho_K^{nk1})^\wedge(\xi) =\cases (1)^{k/2}\pi(n  k  1)A_\xi^{(k)}(0),&\text{if $k$ is even,} Reviewed by Serguey M.Pokas Keywords: convex body; star body; BusemannPetty problem; intersection body; Fourier transform; Radon transform; convexity; parallel section Classification (MSC2000): 52A20 46B07 42B10 Full text of the article:
Electronic fulltext finalized on: 8 Sep 2001. This page was last modified: 21 Jan 2002.
© 2001 Johns Hopkins University Press
