Abstract
Let ℱ be a family of compact starshaped sets in the plane. If every three and every two members of ℱ have a union which is connected and simply connected, then ∩{F:F in ℱ} is simply connected and nonempty. Of course, if every three and every two members of ℱ have a starshaped union, the same result holds.
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Breen, Marilyn, ‘Starshaped unions and nonempty intersections of convex sets in R d,’ Proc. Amer. Math. Soc. 108 (1990), 817–820.
Danzer, Ludwig, Grünbaum, Branko, and Klee, Victor, ‘Helly's theorem and its relatives,’ Convexity, Proc. Symp. Pure Math., Vol. 7, Amer. Math. Soc., Providence, RI, 1962, pp. 101–180.
Kołodziejczyk, Krzysztof, ‘On starshapedness of the union of closed sets in R n,’ Colloq. Math. 53 (1987), 193–197.
Krasnosel'skii, M. A., ‘Sur un critère pour qu'un domaine soit étoilé,’ Mat. Sb. (61) 19 (1946), 309–310.
Lay, Steven R., Convex Sets and Their Applications, Wiley, New York, 1982.
Molnár, J., ‘Über den zweidimensionalen topologischen Satz von Helly,’ Mat. Lapok 8 (1957), 108–114.
Molnár, J., ‘Über eine Verallgemeinerung anf die Kugelfläche eines topologischen Satzes von Helly,’ Acta Math. Acad. Sci. Hungar. 7 (1956), 107–108.
Valentine, F. A., Convex Sets, McGraw-Hill, New York, 1964.
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Supported in part by NSF grants DMS-8705336, DMS-8908717 and by a Senior Faculty Summer Research Fellowship, Research Council, University of Oklahoma.
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Breen, M. An intersection property for starshaped sets in the plane. Geom Dedicata 37, 317–326 (1991). https://doi.org/10.1007/BF00181408
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DOI: https://doi.org/10.1007/BF00181408