Convex body, Intersection, Dilatate, Brunn-Minkowski inequality, Isoperimetric inequality, Symmetral, Ball, Sphere
We initiate a systematic investigation into the nature of the function ∝K(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 ∂(ρ L) 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 S2, 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.
Transactions of the American Mathematical Society
Required Publisher's Statement
First published in Transactions of the American Mathematical Society in Volume 364, Number 3, 2012, published by the American Mathematical Society
Campi, Stefano; Gardner, Richard J.; and Gronchi, Paolo, "Intersections of Dilatates of Convex Bodies" (2012). Mathematics. 29.