A positive solution to the Busemann - Petty problem in R 4

The problem is trivially positive in R2. However, a surprising negative answer for n ≥ 12 was given by Larman and Rogers [LR] in 1975. Subsequently, a series of contributions were made to reduce the dimensions to n ≥ 5 by a number of authors (see [Ba], [Bo], [G2], [Gi], [Pa], and [Z1]). That is, the problem has a negative answer for n ≥ 5. See [G3] for a detailed description. It was proved by Gardner [G1] that the problem has a positive answer for n = 3. The case of n = 4 was considered in [Z1]. But the answer to this case in [Z1] is not correct. This paper presents the correct solution, namely, the Busemann-Petty problem has a positive solution in R4, which, together with results of other cases, brings the Busemann-Petty problem to a conclusion. A key step to the solution of the Busemann-Petty problem is the discovery of the relation of the problem to intersection bodies by Lutwak [Lu]. An originsymmetric convex body K in R is called an intersection body if its radial function ρK is the spherical Radon transform of a nonnegative measure μ on the unit sphere Sn−1. The value of the radial function of K, ρK(u), in the direction u ∈ Sn−1, is defined as the distance from the center of K to its boundary in that direction. When μ is a positive continuous function, K is