  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.
