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Starshapedness vs. Convexity

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Abstract

Starshapedness is a generalization of convexity. A set C is convex if \({\forall x\in C}\) and \({\forall y\in C}\) the segment \({[x:y]\subset C}\). On the other hand, a set S is starshaped if \({\exists y\in S}\) such that \({\forall x\in S}\) the segment \({[x:y]\subset S}\). Due to these closely related definitions, convex and starshaped sets have many similarities, but there are also some striking differences. In this paper we continue our studies of such similarities and differences. Our main goal is to get characterizations of starshapedness and, further on, to describe a starshaped set and its kernel by means of cones included in its complement.

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Authors and Affiliations

  1. Universidad de Luján, Buenos Aires, Argentina

    Guillermo Hansen

  2. Fakultät für Mathematik, Technishe Universität Chemnitz, 09107, Chemnitz, Germany

    Horst Martini

Authors
  1. Guillermo Hansen
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  2. Horst Martini
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Correspondence to Horst Martini.

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To the memory of my friend and teacher Fausto A. Toranzos Toujours la même chose!

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Hansen, G., Martini, H. Starshapedness vs. Convexity. Results. Math. 59, 185–197 (2011). https://doi.org/10.1007/s00025-010-0079-4

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  • DOI: https://doi.org/10.1007/s00025-010-0079-4

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