On a Generalization of the Busemann-Petty Problem

Summary

The generalized Busemann-Petty problem asks: If K and L are origin-symmetric convex bodies in ℝⁿ , and the volume of K ∩ H is smaller than the volume of L ∩ H for every i-dimensional subspace H, 1 < i < n, does it follow that the volume of K is smaller than the volume of L? The hyperplane case i = n -1 is known as the Busemann-Petty problem. It has a negative answer when n > 4, and has a positive answer when n = 3,4. This paper gives a negative answer to· the generalized Busemann-Petty problem for 3 < i < n in the stronger sense that the integer i is not fixed. For the 2-dimensional case i = 2, it is proved that the problem has a positive answer when L is a ball and K is close to L.

1. Introduction

Denote by VOl_i (.) the _i-dimensional Lebesgue measure, and denote by G_i,n the Grassmann manifold of i-dimensional subspaces ofℝ ⁿ. The generalized Busemann-Petty problem asks: G BP. If K and L are origin-symmetric convex bodies in ℝ ⁿ, is there the implication

The case of i = 1 is trivially true. The hyperplane case i = n - 1 is well-known as the Busemann-Petty problem (see [BP] and [Bu]). Many authors contributed to the solution of the Busemann-Petty problem (see [Ba] [Bo] [Gl] [Gia] [Gie] [GR] [Ha] [Lu] [LR] [Pal [ZI]). The problem has a negative answer when n > 4 (see [Gl], [Pal and [Z2]) , and it has a positive answer when n = 3,4 (see [G2] and [Z4]).