Affine diameters of convex bodies

Abstract:

We prove sharp inequalities for the average number of affine diameters through the points of a convex body $K$ in ${\mathbb R}^n$. These inequalities hold if $K$ is a polytope or of dimension two. An example shows that the proof given in the latter case does not extend to higher dimensions. The example also demonstrates that for $n\ge 3$ there exist norms and convex bodies $K\subset \mathbb {R}^n$ such that the metric projection on $K$ with respect to the metric defined by the given norm is well defined but not a Lipschitz map, which is in striking contrast to the planar or the Euclidean case.