A familiär relation links the densities that result for the intersection of a convex body and straight lines under uniform isotropic randomness with those that result under weighted randomness. An extension of this relation to the intersection of more general domains is utilized to obtain the variance of the n-dimensional measure of the intersection of two bodies under uniform isotropic randomn...

In this paper we show that there is no local equatorial characterization of zonoids in odd dimensions. This gives a negative answer to the conjecture posed by W. Weil in 1977 and shows that the local equatorial characterization of zonoids may be given only in even dimensions. In addition we prove a similar result for intersection bodies and show that there is no local characterization of these ...

We prove new Alexandrov-Fenchel type inequalities and new affine isoperimetric inequalities for mixed p-affine surface areas. We introduce a new class of bodies, the illumination surface bodies, and establish some of their properties. We show, for instance, that they are not necessarily convex. We give geometric interpretations of Lp affine surface areas, mixed p-affine surface areas and other ...

The main purposes of this paper are to establish some new Brunn– Minkowski inequalities for width-integrals of mixed projection bodies and affine surface area of mixed bodies, together with their inverse forms.

Mahler’s conjecture predicts a sharp lower bound on the volume of the polar of a convex body in terms of its volume. We confirm the conjecture for convex bodies with many hyperplane symmetries in the following sense: their hyperplanes of symmetries have a one-point intersection. Moreover, we obtain improved sharp lower bounds for classes of convex bodies which are invariant by certain reflectio...