In the mathematical field of convex geometry, the Busemann–Petty problem, introduced by Herbert Busemann and Clinton Myers Petty (1956, problem 1), asks whether it is true that a symmetric convex body with larger central hyperplane sections has larger volume.
Unexpectedly at the time, Larman and Claude Ambrose Rogers (1975) showed that the Busemann–Petty problem has a negative solution in dimensions at least 12, and this bound was reduced to dimensions at least 5 by several other authors.
Ball (1988) pointed out a particularly simple counterexample: all sections of the unit volume cube have measure at most √2, while in dimensions at least 10 all central sections of the unit volume ball have measure at least √2.
Zhang (1994) claimed incorrectly that the unit cube in R4 is not an intersection body, which would have implied that the Busemann–Petty problem has a negative solution if the dimension is at least 4.
Richard J. Gardner, A. Koldobsky, and T. Schlumprecht (1999) gave a uniform solution for all dimensions.