INFINITESIMAL ASPECTS OF THE BUSEMANN-PETTY PROBLEM ERIC L. GRINBERG AND IGOR RIVIN 0. Introduction In an old paper [6] H. Busemann and C. Petty pose the following problem. Does there exist a pair of convex, origin-symmetric bodies B x and B 2 in R n so that for every hyperplane H through the origin we have while We shall refer to this phenomenon as inequality reversal. There has been considerable progress made in understanding this problem, but it remains open in its full generality. The hypotheses of convexity and origin-centredness above were imposed after Busemann [5] found fairly simple counterexamples in cases where at least one of these conditions is removed. Below we exhibit even simpler examples. The initial motivation for this problem probably came from an old inequality of Busemann [4]: Vol^M)"" 1 . (1) HettP" Here the integration is over the set RP n ~ x of hyperplanes H through the origin in R n , dH is the normalized rotation-invariant measure on RP n ~\ Vol n _ x (M n H) denotes the cross-sectional area of the body M in H, and Vol n (M) is the Euclidean volume of M. The constant c n is chosen so that (1) is an equality if M is the unit ball centred at the origin. The inequality is valid for M a measurable body in R n (Busemann proved (1) for convex bodies; the extension to the measurable category was given by Petty in [13]). The «th root of the left-hand side of (1) is called the 1st dual affine Quermassintegral of the body M and is denoted by O X (M) (see Lutwak [11, 12]). Busemann showed that if equality holds in (1) and M is convex, then M must be an origin-centred ellipsoid. A ^-dimensional version of the equality portion of (1) was given by Furstenberg and Tzkoni [7] for M an origin-centred ellipsoid: (M n H)} n dH = Vol n (Mf. HeG(k,n) Here the integration is over G(k, n), the Grassmann manifold of ^-dimensional vector subspaces of R n . The ^-dimensional version of the mequality along with a Received 17 April 1989; revised 16 March 1990. 1980 Mathematics Subject Classification 52A40, 53C65. First author supported in part by a grant from the National Science Foundation; second author supported by a contract from the Defense Advanced Research Projects Agency. Bull. London Math. Soc. 22 (1990) 478-484