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Ball and spindle convexity with respect to a convex body

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Abstract

Let \({C \subset \mathbb{R}^n}\) be a convex body. We introduce two notions of convexity associated to C. A set K is C-ball convex if it is the intersection of translates of C, or it is either \({\emptyset}\) , or \({\mathbb{R}^n}\) . The C-ball convex hull of two points is called a C-spindle. K is C-spindle convex if it contains the C-spindle of any pair of its points. We investigate how some fundamental properties of conventional convex sets can be adapted to C-spindle convex and C-ball convex sets. We study separation properties and Carathéodory numbers of these two convexity structures. We investigate the basic properties of arc-distance, a quantity defined by a centrally symmetric planar disc C, which is the length of an arc of a translate of C, measured in the C-norm that connects two points. Then we characterize those n-dimensional convex bodies C for which every C-ball convex set is the C-ball convex hull of finitely many points. Finally, we obtain a stability result concerning covering numbers of some C-ball convex sets, and diametrically maximal sets in n-dimensional Minkowski spaces.

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Authors and Affiliations

  1. Department of Geometry, Budapest University of Technology and Economics, Egry József u. 1, Budapest, 1111, Hungary

    Zsolt Lángi

  2. Department of Geometry, Eötvös University, Pázmány Péter s. 1/c, Budapest, 1117, Hungary

    Márton Naszódi

  3. Department of Geometry and Informatics, Ybl Faculty of Architecture and Civil Engineering Szent István University, Thököly út 74, Budapest, 1146, Hungary

    István Talata

Authors
  1. Zsolt Lángi
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  2. Márton Naszódi
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  3. István Talata
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Correspondence to Zsolt Lángi.

Additional information

Z. Lángi was supported by the János Bolyai Research Scholarship of the Hungarian Academy of Sciences. M. Naszódi was partially supported by the Hung. Nat. Sci. Found. (OTKA) Grants: K72537 and PD104744. I. Talata was partially supported by the Hung. Nat. Sci. Found. (OTKA) Grant no. K68398.

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Lángi, Z., Naszódi, M. & Talata, I. Ball and spindle convexity with respect to a convex body. Aequat. Math. 85, 41–67 (2013). https://doi.org/10.1007/s00010-012-0160-z

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  • DOI: https://doi.org/10.1007/s00010-012-0160-z

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