DC Field | Value | Language |
---|---|---|

dc.contributor.author | Makai, Endre | en |

dc.contributor.author | Vrećica, Siniša | en |

dc.contributor.author | Živaljević, Rade | en |

dc.date.accessioned | 2020-04-12T18:03:59Z | - |

dc.date.available | 2020-04-12T18:03:59Z | - |

dc.date.issued | 2001-01-01 | en |

dc.identifier.issn | 01795376 | en |

dc.identifier.uri | http://researchrepository.mi.sanu.ac.rs/handle/123456789/306 | - |

dc.description.abstract | Let 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.publisher | Springer Link | - |

dc.relation.ispartof | Discrete and Computational Geometry | en |

dc.title | Plane sections of convex bodies of maximal volume | en |

dc.type | Article | en |

dc.identifier.doi | 10.1007/s004540010070 | en |

dc.identifier.scopus | 2-s2.0-0040481369 | en |

dc.contributor.affiliation | Mathematical Institute of the Serbian Academy of Sciences and Arts | - |

dc.relation.firstpage | 33 | en |

dc.relation.lastpage | 49 | en |

dc.relation.issue | 1 | en |

dc.relation.volume | 25 | en |

dc.description.rank | M21 | - |

item.openairetype | Article | - |

item.cerifentitytype | Publications | - |

item.fulltext | No Fulltext | - |

item.openairecristype | http://purl.org/coar/resource_type/c_18cf | - |

item.grantfulltext | none | - |

crisitem.author.orcid | 0000-0001-9801-8839 | - |

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