  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 $\omega \in S^{d-1}$, we 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.
