Intersection bodies that are not polar zonoids: A flat top condition in dimensions four and six

TL;DR

This paper establishes geometric conditions in four and six dimensions under which the intersection body of a convex body of revolution is not a polar zonoid, providing explicit examples of such bodies.

Contribution

It introduces new geometric criteria for intersection bodies in specific dimensions, demonstrating cases where they are not polar zonoids, expanding understanding of convex geometric structures.

Findings

Identified conditions preventing intersection bodies from being polar zonoids in 4D and 6D.

Constructed explicit examples of intersection bodies that are not polar zonoids.

Enhanced the classification of convex bodies based on their intersection and zonoid properties.

Abstract

We find general geometric conditions on a convex body of revolution K, in dimensions four and six, so that its intersection body IK is not a polar zonoid. We exhibit several examples of intersection bodies which are are not polar zonoids.