Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

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Structure of closed finitely starshaped sets
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by Mabel A. Rodriguez and Fausto A. Toranzos
Proc. Amer. Math. Soc. 128 (2000), 1433-1441
DOI: https://doi.org/10.1090/S0002-9939-00-05620-3
Published electronically: February 7, 2000
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Abstract:

A set $S$ is finitely starshaped if any finite subset of $S$ is totally visible from some point of $S$. It is well known that in a finite-dimensional linear space, a closed finitely starshaped set which is not starshaped must be unbounded. It is proved here that such a set must admit at least one direction of recession. This fact clarifies the structure of such sets and allows the study of properties of their visibility elements, well known in the case of starshaped sets. A characterization of planar finitely starshaped sets by means of its convex components is obtained. Some plausible conjectures are disproved by means of counterexamples.
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Bibliographic Information
  • Mabel A. Rodriguez
  • Affiliation: Instituto de Desarrollo Humano, Universidad Nacional de General Sarmiento, Roca 850, (1663) San Miguel, Argentina
  • Email: mrodri@ungs.edu.ar
  • Fausto A. Toranzos
  • Affiliation: Departamento de Matemática, Universidad de Buenos Aires, Ciudad Universitaria, (1428) Buenos Aires, Argentina
  • Email: fautor@dm.uba.ar
  • Received by editor(s): May 13, 1998
  • Published electronically: February 7, 2000
  • Additional Notes: This paper was written as part of Argentine Research Project 01/TX38
  • Communicated by: Christopher Croke
  • © Copyright 2000 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 128 (2000), 1433-1441
  • MSC (2000): Primary 52A30, 52A35
  • DOI: https://doi.org/10.1090/S0002-9939-00-05620-3
  • MathSciNet review: 1706993