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dc.contributor.authorMakai, Endreen
dc.contributor.authorVrećica, Sinišaen
dc.contributor.authorŽivaljević, Radeen
dc.date.accessioned2020-04-12T18:03:59Z-
dc.date.available2020-04-12T18:03:59Z-
dc.date.issued2001-01-01en
dc.identifier.issn01795376en
dc.identifier.urihttp://researchrepository.mi.sanu.ac.rs/handle/123456789/306-
dc.description.abstractLet K = {K0, . . . , Kk} be a family of convex bodies in Rn, 1 ≤ k ≤ n - 1. We prove, generalizing results from [9], [10], [13], and [14], that there always exists an affine k-dimensional plane Ak ⊆ Rn, called a common maximal k-transversal of K, such that, for each i ∈ {0, . . . , k} and each x ∈ Rn, Vk(Ki ∩ Ak) ≥ Vk(Ki ∩ (Ak + x)), where Vk is the k-dimensional Lebesgue measure in Ak and Ak + x. Given a family K = {Ki}li=0 of convex bodies in Rn, l < k, the set Ck(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 Ck(K) cannot be included in a v-dimensional C1 submanifold (or more generally in an (Hv, v)-rectifiable, Hv-measurable subset) of the affine Grassmannian AGrn,k of all affine k-dimensional planes of Rn, of O(n + 1)-invariant v-dimensional (Hausdorff) measure less than some positive constant Cn,k,l, where v = (k - l)(n - k). As usual, the "affine" Grassmannian AGrn,k is viewed as a subspace of the Grassmannian Grn+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 θ ∈ Hn-k(Gn+1,k+1; Z2) such that the class θl+1 is concentrated in an arbitrarily small neighborhood of Ck(K). As an immediate consequence we deduce that the Lyusternik-Shnirel'man category of the space Ck(K) relative to Grn+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 Grn+1,k+1 has to be large also in a measure theoretic sense.en
dc.publisherSpringer Link-
dc.relation.ispartofDiscrete and Computational Geometryen
dc.titlePlane sections of convex bodies of maximal volumeen
dc.typeArticleen
dc.identifier.doi10.1007/s004540010070en
dc.identifier.scopus2-s2.0-0040481369en
dc.contributor.affiliationMathematical Institute of the Serbian Academy of Sciences and Arts-
dc.relation.firstpage33en
dc.relation.lastpage49en
dc.relation.issue1en
dc.relation.volume25en
dc.description.rankM21-
item.openairetypeArticle-
item.cerifentitytypePublications-
item.fulltextNo Fulltext-
item.openairecristypehttp://purl.org/coar/resource_type/c_18cf-
item.grantfulltextnone-
crisitem.author.orcid0000-0001-9801-8839-
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