Intersections of dilatates of convex bodies

Abstract:

We initiate a systematic investigation into the nature of the function $\alpha _K(L,\rho )$ that gives the volume of the intersection of one convex body $K$ in $\mathbb {R}^n$ and a dilatate $\rho L$ of another convex body $L$ in $\mathbb {R}^n$, as well as the function $\eta _K(L,\rho )$ that gives the $(n-1)$-dimensional Hausdorff measure of the intersection of $K$ and the boundary $\partial (\rho L)$ of $\rho L$. The focus is on the concavity properties of $\alpha _K(L,\rho )$. 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 $\alpha _K(L,\rho )$ 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 $S^2$, 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.