Positivity 3: 95–100, 1999.
© 1999 Kluwer Academic Publishers. Printed in the Netherlands.
A Short Solution to the Busemann-Petty Problem
F. BARTHE, M. FRADELIZI and B. MAUREY
Equipe d’Analyse et Mathématiques Appliquées, Université de Marne la Vallée, Boulevard
Descartes, Cité Descartes, Champs sur Marne, 77454 Marne la Vallée Cedex 2, France
(Received: 3 September 1998; Accepted: 3 September 1998)
Abstract. A uniﬁed analytic solution to the Busemann-Petty problem was recently found by Gard-
ner, Koldobsky and Schlumprecht. We give an elementary proof of their formulas for the inverse
Radon transform of the radial function ρ
of an origin-symmetric star body K.
Mathematics Subject Classiﬁcations (1991): 44A12, 52A20, 52A38
Key words: convex body, star body, Busemann-Petty problem, Radon transform
The 1956 Busemann-Petty problem asks the following question : suppose that K
and L are origin-symmetric convex bodies in
(K ∩ H)
(L ∩ H)
for every hyperplane H containing the origin; does it follow that
The problem has a long and dramatic history. A negative answer to the problem
5 was established in a series of papers by Larman and Rogers  (for
12), Ball  (n
10), Giannopoulos  and Bourgain  (independently;
7), Gardner  and Papadimitrakis  (independently; n
5). Gardner 
proved that the answer to the Busemann-Petty problem is afﬁrmative when n = 3.
A negative answer in the case n = 4 was claimed in 1994, but three years later the
main argument of that proof was shown to be wrong (for details, see Koldobsky
). After that, Zhang  showed that the answer is afﬁrmative when n = 4, and,
a little later, a uniﬁed solution to the problem was given by Gardner, Koldobsky
and Schlumprecht in .
The principal objective of this paper is to present an elementary proof for the
main positive result, namely the solution of the Busemann-Petty problem in four
dimensions. Gardner, Koldobsky and Schlumprecht proved in  that the radial
of a smooth symmetric convex body K in
is the Radon transform
of an explicit non negative function (see below); according to the 1988 result of