#### Document Type

Article

#### Publication Date

3-2012

#### 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

**R**

^{n}and a dilatate

**ρ**

*L*of another convex body

*L*in

**R**

^{n}, 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 ∂(

**ρ**

*L*) of

**ρ**

*L*. The focus is on the concavity properties of

**α**

_{K}(

*L,*). Of particular interest is the case when

**ρ***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,*) between dimension 2 and dimensions 3 or higher. When

**ρ***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.

#### 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.* Paper 29.

http://cedar.wwu.edu/math_facpubs/29