Intersections of Dilatates of Convex Bodies

We initiate a systematic investigation into the nature of the function αK (L, ρ) that gives the volume of the intersection of one convex body K in Rn and a dilatate ρL of another convex body L in Rn, as well as the function ηK (L, ρ) that gives the (n − 1)-dimensional Hausdorff measure of the intersection of K and the boundary ∂(ρL) of ρL. The focus is on the concavity properties of αK (L, ρ). Of particular interest is the case when K and L are symmetric with respect to the origin. In this situation, there is an interesting change in the concavity properties of αK (L, ρ) between dimension 2 and dimensions 3 or higher. When L is the unit ball, an important special case with connections to E. Lutwak’s dual Brunn-Minkowski theory, we prove that this change occurs between dimension 2 and dimensions 4 or higher, and conjecture that it occurs between dimension 3 and dimension 4. We also establish an isoperimetric inequality with equality condition for subsets of equatorial zones in the sphere S , and apply this and the Brunn-Minkowski inequality in the sphere to obtain results related to this conjecture, as well as to the properties of a new type of symmetral of a convex body, which we call the equatorial symmetral.