Document Type
Article
Publication Date
3-2012
Keywords
Convex body, Intersection, Dilatate, Brunn-Minkowski inequality, Isoperimetric inequality, Symmetral, Ball, Sphere
Abstract
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.
Publication Title
Transactions of the American Mathematical Society
Volume
364
Issue
3
First Page
1193
Last Page
1210
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
Recommended Citation
Campi, Stefano; Gardner, Richard J.; and Gronchi, Paolo, "Intersections of Dilatates of Convex Bodies" (2012). Mathematics Faculty Publications. 29.
https://cedar.wwu.edu/math_facpubs/29
Subjects - Topical (LCSH)
Convex bodies; Intersection theory (Mathematics)
Genre/Form
articles
Type
Text
Rights
Copying of this document in whole or in part is allowable only for scholarly purposes. It is understood, however, that any copying or publication of this document for commercial purposes, or for financial gain, shall not be allowed without the author’s written permission.
Language
English
Format
application/pdf
