Abstract
Let \(\mathcal{K} = \{ K_0 , \ldots ,K_k \}\) be a family of convex bodies in R 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 k ∈ R n, called a common maximal k-transversal of \(\mathcal{K}\), such that, for each i ∈ {0, ..., k} and each x ∈ R n,
where V k is the k-dimensional Lebesgue measure in A k and A k + x. Given a family \(\mathcal{K} = \{ K_i \} _{i = 0}^l\) of convex bodies in R n, l < k, the set \(C_k (\mathcal{K})\) of all common maximal k-transversals of \(\mathcal{K}\) is not only nonempty but has to be “large” both from the measure theoretic and the topological point of view. It is shown that \(C_k (\mathcal{K})\) cannot be included in a ν-dimensional C 1 submanifold (or more generally in an \((\mathcal{H}^\nu ,\nu )\)-rectifiable, \(\mathcal{H}^\nu\)-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 \(C_k (\mathcal{K})\). As an immediate consequence we deduce that the Lyusternik-Shnirel’man category of the space \(C_k (\mathcal{K})\) 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.
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Received May 22, 1998, and in revised form March 27, 2000. Online publication September 22, 2000.
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Makai, E., Vrecica, S. & Živaljevic, R. Plane Sections of Convex Bodies of Maximal Volume . Discrete Comput Geom 25, 33–49 (2001). https://doi.org/10.1007/s004540010070
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DOI: https://doi.org/10.1007/s004540010070