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On the number of vertices of projective polytopes

  • Identification data

    Identifier: imarina:9295555
    Authors:
    Garcia-Colin, NMontejano, LPAlfonsin, JLR
    Abstract:
    Let X be a set of n points in Rd$\mathbb {R}<^>d$ in general position. What is the maximum number of vertices that conv(T(X))$\mathsf {conv}(T(X))$ can have among all the possible permissible projective transformations T? In this paper, we investigate this and other related questions. After presenting several upper bounds, obtained by using oriented matroid machinery, we study a closely related problem (via Gale transforms) concerning the maximal number of minimal Radon partitions of a set of points. The latter led us to a result supporting a positive answer to a question of Pach and Szegedy asking whether balanced 2-colorings of points in the plane maximize the number of induced multicolored Radon partitions. We also discuss a related problem concerning the size of topes in arrangements of hyperplanes as well as a tolerance-type problem of finite sets.
  • Others:

    Author, as appears in the article.: Garcia-Colin, N; Montejano, LP; Alfonsin, JLR
    Department: Enginyeria Informàtica i Matemàtiques
    URV's Author/s: Montejano Cantoral, Luis Pedro
    Keywords: Proof Cells Arrangements
    Abstract: Let X be a set of n points in Rd$\mathbb {R}<^>d$ in general position. What is the maximum number of vertices that conv(T(X))$\mathsf {conv}(T(X))$ can have among all the possible permissible projective transformations T? In this paper, we investigate this and other related questions. After presenting several upper bounds, obtained by using oriented matroid machinery, we study a closely related problem (via Gale transforms) concerning the maximal number of minimal Radon partitions of a set of points. The latter led us to a result supporting a positive answer to a question of Pach and Szegedy asking whether balanced 2-colorings of points in the plane maximize the number of induced multicolored Radon partitions. We also discuss a related problem concerning the size of topes in arrangements of hyperplanes as well as a tolerance-type problem of finite sets.
    Thematic Areas: Mathematics, applied Mathematics (miscellaneous) Mathematics (all) Mathematics General mathematics
    licence for use: https://creativecommons.org/licenses/by/3.0/es/
    Author's mail: luispedro.montejano@urv.cat
    Record's date: 2024-08-03
    Papper version: info:eu-repo/semantics/publishedVersion
    Link to the original source: https://londmathsoc.onlinelibrary.wiley.com/doi/full/10.1112/mtk.12193
    Licence document URL: https://repositori.urv.cat/ca/proteccio-de-dades/
    Papper original source: Mathematika. 69 (2): 535-561
    APA: Garcia-Colin, N; Montejano, LP; Alfonsin, JLR (2023). On the number of vertices of projective polytopes. Mathematika, 69(2), 535-561. DOI: 10.1112/mtk.12193
    Article's DOI: 10.1112/mtk.12193
    Entity: Universitat Rovira i Virgili
    Journal publication year: 2023
    Publication Type: Journal Publications
  • Keywords:

    Mathematics,Mathematics (Miscellaneous),Mathematics, Applied
    Proof
    Cells
    Arrangements
    Mathematics, applied
    Mathematics (miscellaneous)
    Mathematics (all)
    Mathematics
    General mathematics
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