Centrally symmetric convex bodies and sections having maximal quermassintegrals

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Let d ≥ 2, and let K ⊂ Rᵈ 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 ω∈ Sᵈ⁻¹, we have that the (d-1)-volume of the intersection of K and an arbitrary hyperplane, with normal ω, 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 (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 C²-perturbations, or C³-perturbations of the Euclidean unit ball, respectively.