ABSTRACT:
The Busemann-Petty problem (posed in 1956) asks the following question.
Suppose that K and L are two origin-symmetric convex bodies in
n-dimensional Euclidean space such that the ((n-1)-dimensional) volume of
each central hyperplane section of K is smaller than the volume of the
corresponding section of L. Does it follow that the (n-dimensional)
volume of K is smaller than the volume of L?
The answer is negative in dimensions 5 and larger (it was proved during
the period from 1975 to 1993 by individual counterexamples lowering the
dimension from 12 to 5). The answer is affirmative for dim=3
(R.J.Gardner, 1994). Until very recently, the answer in the case dim=4
was considered to be negative, but several months ago G. Zhang revised the
solution and showed that the opposite is true.
We present a new approach to the Busemann-Petty problem, based on the use
of the Fourier transform, which initiated the revision of the case dim=4,
provided a large class of bodies K for which the answer is positive for
every choice of L with larger sections, and put the production of
counterexamples in dim=5 and higher on the industrial level. This approach
also leads to a new simple solution to the problem (in all dimensions),
and explains why the transition occurs between dim=4 and dim=5.