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By Milman Emanuel (il-weiz


(2001m:52007)], a centrally-symmetric star-body K is said to be a k-intersection body of a starbody L if Vol(K ∩ H ⊥ ) = Vol(L ∩ H) for any H ∈ G(n, n − k). Further, K is said to be a k-intersection body if it is the limit in the radial metric dr of k-intersection bodies {Ki} of star-bodies {Li}. The main result of this work is that for n ≥ 4 and 2 ≤ k ≤ n − 2, there exists an infinite smooth centrally-symmetric body of revolution K which is a k-intersection body, but not a k-Busemann-Petty body, answering in the negative the corresponding question of Koldobsky. This counter-example can be reformulated in various ways, including the assertion that for n ≥ 4 there exist nontrivial n-dimensional spaces which embed in Lp for −2 ≥ p ≥ n − 2. It also implies the existence of nontrivial non-negative functions in the range of the spherical Radon transform. Reviewed by Aris Daniilidi

Year: 2013
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