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Structure of closed finitely starshaped sets

Authors: Mabel A. Rodriguez and Fausto A. Toranzos
Journal: Proc. Amer. Math. Soc. 128 (2000), 1433-1441
MSC (2000): Primary 52A30, 52A35
Published electronically: February 7, 2000
MathSciNet review: 1706993
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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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Additional Information

Mabel A. Rodriguez
Affiliation: Instituto de Desarrollo Humano, Universidad Nacional de General Sarmiento, Roca 850, (1663) San Miguel, Argentina

Fausto A. Toranzos
Affiliation: Departamento de Matemática, Universidad de Buenos Aires, Ciudad Universitaria, (1428) Buenos Aires, Argentina

Keywords: Finitely starshaped sets, cone of recession, convex components
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
Article copyright: © Copyright 2000 American Mathematical Society