Intersection Bodies in R4

✍ Scribed by Alexander Koldobsky

• Book ID: 102562478
• Publisher: Elsevier Science
• Year: 1998
• Tongue: English
• Weight: 251 KB
• Volume: 136
• Category: Article
• ISSN: 0001-8708
• DOI: 10.1006/aima.1998.1718

✦ Synopsis

Intersection bodies were introduced in 1988 by Lutwak, who found a close connection between those bodies and the well-known 1956 Busemann Petty problem, which asks whether origin symmetric convex bodies with larger central hyperplane sections also have greater volume. The author has recently shown that an origin symmetric star body K is an intersection body if and only if the function &x& &1 K is a positive definite distribution, where &x& K =min[a 0: x # aK]. We use this result to prove that the unit balls of the spaces l 4 q , 2<q , are intersection bodies. For n 5 the unit balls of the spaces l n q , 2<q , are not intersection bodies. The technique of this paper allows to find precise expressions for the generating measures (signed measures) of these bodies. The result that the unit cube in R 4 is an intersection body shows that the negative solution to the Busemann Petty problem for n=4 in [20] was incorrect.

1998 Academic Press

where u [ Rf (u)= 0 & u = f (x) dx, u # 0, is the spherical Radon transform defined for every continuous function f on 0.