In this small survey we consider the volume product, and sketch some of the best upper and lower estimates known up to now, based on our paper [BMMR]. The author thanks the organizers of the conference in Jurata, March 2010, for their kind invitation
, and the excellent atmosphere there. This paper is based on the talk of the author on that conference.
Let $d ge 2$, and let $K subset {Bbb{R}}^d$ be a convex body containing the origin $0$ in its interior. In a previous paper we have proved the following. The body $K$ is $0$-symmetric if and only if the following holds. For each $omega in S^{d-1}$, w
e have that the $(d-1)$-volume of the intersection of $K$ and an arbitrary hyperplane, with normal $omega$, attains its maximum if the hyperplane contains $0$. An analogous theorem, for $1$-dimensional sections and $1$-volumes, has been proved long ago by Hammer (cite{H}). In this paper we deal with the ($(d-2)$-dimensional) surface area, or with lower dimensional quermassintegrals of these intersections, and prove an analogous, but local theorem, for small $C^2$-perturbations, or $C^3$-perturbations of the Euclidean unit ball, respectively.
