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Separoids, Their Categories and a Hadwiger-Type Theorem for Transversals

Separoids, Their Categories and a Hadwiger-Type Theorem for Transversals,10.1007/s00454-001-0075-2,Discrete & Computational Geometry,Jorge L. Arocha,J

Separoids, Their Categories and a Hadwiger-Type Theorem for Transversals   (Citations: 12)
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In this paper we study the topology of transversals to a family of convex sets as a subset of a Grassmanian manifold. This topology seems to be ruled by a combinatorial structure which we call a separoid. With these combinatorial objects and the topological notion of virtual transversal we prove a Borsuk-Ulam-type theorem which has as a corollary a generalization of Hadwiger's theorem.
Journal: Discrete & Computational Geometry - DCG , vol. 27, no. 3, pp. 377-385, 2002
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    • ...[2]). This is a key ingredient in [1, 3, 6]...
    • ...Observe that the separoid of a family of convex sets does not distinguish if three sets have a point in common, or if they simply intersect pairwise (see Fig. 1)—this phenomenon motivated us to introduce the notion of a virtual � -transversal in [1] and of a virtual Tverberg partition in [3]...

    Ricardo Strausz. Hyperseparoids: A Representation Theorem

    • ...Several results on transversals, similar to the results of this paper, can be found in [1, 4, 5, 11]...

    L. Montejanoet al. Topological transversals to a family of convex sets

    • ...More precisely, and using the simple terminology of separoids [3]: for every index partition α ∪ β = I with α ∩ β = ∅, there exists a hyperplane L C Rm that separates {qi}i∈α from {qi}i∈β if and only if there exists a hyperplane L 0 C Y that separates {yi}i∈α...

    J. L. Arochaet al. Flat transversals to flats and convex sets of a fixed dimension

    • ...More recently, the concept of a separoid was introduced [1, 3, 12, 13, 16, 17, 18] as a new attempt in this direction that is instead based on Radon’s theorem (1921) [14]...
    • ...Conversely, as proved by Arocha et al. [1], every (abstract) separoid can be represented in such a way by a family of convex sets in some Euclidean space...
    • ...Arocha et al. [1] proved a Hadwiger-type theorem that, supposing the existence of a monomorphism ‘from the left’ :P ! S, concludes the existence of a virtual ‘-transversal...
    • ...Given two separoids S and P, a function ’:S ! P is a morphism if the preimage of separations are separations (see [1, 17] for several important examples of morphisms); that is, for all , P,...
    • ...On the other hand, if we suppose —as did Goodman & Pollack [7] and Arocha et al. [1]— that the monomorphism comes ‘from the left’ µ:P ! S, then such a hypothesis is not needed and the argument is much simpler (see the proof of Lemma 1). Observe that Figure 3 also shows that the existence of a virtual line does not imply the existence of the corresponding chromomorphism...
    • ...But then, a | [[1]], where [[1]] denotes the chromatic class of 1; this is a contradiction...
    • ...But then, a | [[1]], where [[1]] denotes the chromatic class of 1; this is a contradiction...

    Juan José Montellano-ballesteroset al. Tverberg-Type Theorems for Separoids

    • ...Conversely, any abstract finite separoid can be represented by a family of convex sets [1]...
    • ...This notion was introduced in [1] to study the topology of hyperplanes transversal to families of convex sets...

    Javier Brachoet al. Two geometric representation theorems for separoids

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