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 η K(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 α K(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 α K(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. © 2011 American Mathematical Society.
Intersections of dilatates of convex bodies / S.Campi; R.J.Gardner; P.Gronchi. - In: TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY. - ISSN 0002-9947. - STAMPA. - 364:(2012), pp. 1193-1210. [10.1090/S0002-9947-2011-05455-3]
Intersections of dilatates of convex bodies
GRONCHI, PAOLO
2012
Abstract
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 η K(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 α K(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 α K(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. © 2011 American Mathematical Society.File | Dimensione | Formato | |
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