Plane Sections of Convex Bodies of Maximal Volume

(ProQuest: ... denotes formulae and/or non-USASCII text omitted; see image) Let K = {K^sub 0^ ,... ,K^sub k^ } be a family of convex bodies in R^sup n^ , 1≤ k≤ n-1 . We prove, generalizing results from [9], [10], [13], and [14], that there always exists an affine k -dimensional plane A^sub k^ (subset, dbl equals) R^sup n^ , called a common maximal k-transversal of K , such that, for each i {0,... ,k} and each x R^sup n^ , ... where V^sub k^ is the k -dimensional Lebesgue measure in A^sub k^ and A^sub k^ +x . Given a family K = {K^sub i^ }^sub i=0^^sup l^ of convex bodies in R^sup n^ , l < k , the set C^sub k^ ( K ) of all common maximal k -transversals of K is not only nonempty but has to be ``large'' both from the measure theoretic and the topological point of view. It is shown that C^sub k^ ( K ) cannot be included in a ν -dimensional C^sup 1^ submanifold (or more generally in an ( H ^sup ν^ , ν) -rectifiable, H ^sup ν^ -measurable subset) of the affine Grassmannian AGr^sub n,k^ of all affine k -dimensional planes of R^sup n^ , of O(n+1) -invariant ν -dimensional (Hausdorff) measure less than some positive constant c^sub n,k,l^ , where ν = (k-l)(n-k) . As usual, the ``affine'' Grassmannian AGr^sub n,k^ is viewed as a subspace of the Grassmannian Gr^sub n+1,k+1^ of all linear (k+1) -dimensional subspaces of R^sup n+1^ . On the topological side we show that there exists a nonzero cohomology class [theta] H^sup n-k^ (G^sub n+1,k+1^ ;Z^sub 2^ ) such that the class [theta]^sup l+1^ is concentrated in an arbitrarily small neighborhood of C^sub k^ ( K ) . As an immediate consequence we deduce that the Lyusternik--Shnirel'man category of the space C^sub k^ ( K ) relative to Gr^sub n+1,k+1^ is ≥ k-l . Finally, we show that there exists a link between these two results by showing that a cohomologically ``big'' subspace of Gr^sub n+1,k+1^ has to be large also in a measure theoretic sense.[PUBLICATION ABSTRACT]