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

TL;DR

This paper proves that in four-dimensional space, if two symmetric convex bodies have smaller central hyperplane slices for one body, then its volume is also smaller, resolving a long-standing open case.

Contribution

The paper provides the first correct proof that the Busemann-Petty problem has a positive answer in R^4, completing the understanding of the problem across dimensions.

Findings

Positive solution to the Busemann-Petty problem in R^4

Clarification of the problem's status in different dimensions

Resolution of a long-standing open question in convex geometry

Abstract

H. Busemann and C. M. Petty posed the following problem in 1956: If K and L are origin-symmetric convex bodies in R^n and for each hyperplane H through the origin the volumes of their central slices satisfy vol(K cap H) < vol(L cap H), does it follow that the volumes of the bodies themselves satisfy vol(K) < vol(L)? The problem is trivially positive in R^2. However, a surprising negative answer for n <= 12 was given by Larman and Rogers in 1975. Subsequently, a series of contributions were made to reduce the dimensions to n >= 5 by a number of authors. That is, the problem has a negative answer for n >= 5. It was proved by Gardner that the problem has a positive answer for n=3. The case of n=4 was considered in [Ann. of Math. (2) 140 (1994), 331-346], but the answer given there is not correct. This paper presents the correct solution, namely, the Busemann-Petty problem has a positive solution in R^4, which, together with results of other cases, brings the Busemann-Petty problem to a conclusion.