Plane sections of convex bodies of maximal volume

Let K = {K-0,..., K-k} be a family of convex bodies in R-n, 1 less than or equal to k less than or equal to n - 1. We prove, generalizing results from [9], [10], [13], and [14], that there always exists an affine k-dimensiona l plane A(k) subset of or equal to R-n, called a common maximal k-transvers al of K, such that, for each i is an element of {0,..., k} and each x is an element of R-n, V-k(K-i boolean AND A(k)) greater than or equal to V-k(K-i boolean AND (A(k ) + x)), where V-k is the k-dimensional Lebesgue measure in A(k) and A(k) + x. Given a family K = {K-i}(i=0)(l) of convex bodies in R-n, l < k, the set C-k(K) Of all common maximal k-transversals of K is not only nonempty but has to b e "large" both from the measure theoretic and the topological point of view . It is shown that C-k(K) cannot be included in a v-dimensional C-1 submani fold(or more generally in an (H-v, v)-rectifiable, H-v-measurable subset) o f the affine Grassmannian AGr(n,k) of all affine k-dimensional planes of R- n, of O(n + 1)-invariant v-dimensional (Hausdorff) measure less than some p ositive constant c(n,k,i), where v = (k - l)(n - k). As usual, the "affine" Grassmannian AGr(n,k) is viewed as a subspace of the Grassmannian Gr(n+1,k +1) of all linear (k + 1)-dimensional subspaces of Rn+1. On the topological side we show that there exists a nonzero cohomology class <theta> is an el ement of Hn-k(G(n+1,k+1); Z(2)) such that the class theta (l+1) is concentr ated in an arbitrarily small neighborhood of C-k(K) AS an immediate consequ ence we deduce that the Lyusternik-Shnirel'man category of the space C-k(K) relative to Gr(n+1,k+1) is greater than or equal to k - l. Finally, we sho w that there exists a link between these two results by showing that a coho mologically "big" subspace of Gr(n+1,k+1) has to be large also in a measure theoretic sense.