Plane Sections of Convex Bodies of Maximal Volume

Abstract

Abstract. Let K = {K 0 , ..., K k } be a family of convex bodies in R n , 1 ≤ k ≤ n−1. We prove, generalizing results from [9], [10], [13], [14], that there always exists an affine k-dimensional plane A k ⊆ R n , called a common maximal k-transversal of K, such that for each i ∈ {0, ..., k} and eachof convex bodies in R n , l < k, the set C k (K) of all common maximal ktransversals of K is not only non-empty but has to be "large" both from the measure theoretic and the topological point of view. It is shown that C k (K) cannot be included in a ν-dimensional C 1 submanifold (or more generally in an (H ν , ν)-rectifiable, H ν -measurable subset) of the affine Grassmannian AGr n,k of all affine k-dimensional planes of R n , of O(n + 1)-invariant ν-dimensional (Hausdorff) measure less than some positive constant c n,k,l , where ν = (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 R n+1 . On the topological side we show that there exists a nonzero cohomology class θ ∈ H n−k (G n+1,k+1 ; Z 2 ) such that the class θ l+1 is concentrated in an arbitrarily small neighborhood of C k (K). As an immediate consequence we deduce that the Lyusternik-Shnirel'man category of the space C k (K) relative to Gr 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 n+1,k+1 has to be large also in a measure theoretic sense.

Топологические Методы В Комбинаторной Геометрии

“…, K k из k+1 выпуклого компакта в R d найдется k-плоскость L, которая является максимальной трансверсалью одновременно для всех K i . В работе [132] приводятся и другие следствия теоремы 81, в частности, характеризующие множество общих максимальных k-трансверсалей для семейства из l k выпуклых компактов.…”