Starshaped unions and nonempty intersections of convex sets in $\textbf {R}^ d$
HTML articles powered by AMS MathViewer
- by Marilyn Breen PDF
- Proc. Amer. Math. Soc. 108 (1990), 817-820 Request permission
Abstract:
Let $\mathcal {G}$ be a nonempty family of compact convex sets in ${R^d},d \geq 1$. Then every subfamily of $\mathcal {G}$ consisting of $d + 1$ or fewer sets has a starshaped union if and only if $\cap \{ G:G{\text { in }}\mathcal {G}\} \ne \emptyset$.References
-
Ludwig Danzer, Branko Grünbaum, and Victor Klee, Helly’s theorem and its relatives, Convexity, Proc. Sympos. Pure Math., Vol. 7, Amer. Math. Soc., Providence, RI, 1962, pp. 101-180.
- V. L. Klee Jr., On certain intersection properties of convex sets, Canad. J. Math. 3 (1951), 272–275. MR 42726, DOI 10.4153/cjm-1951-031-2
- Krzysztof Kołodziejczyk, On starshapedness of the union of closed sets in $\textbf {R}^n$, Colloq. Math. 53 (1987), no. 2, 193–197. MR 924062, DOI 10.4064/cm-53-2-193-197
- M. Krasnosselsky, Sur un critère pour qu’un domaine soit étoilé, Rec. Math. [Mat. Sbornik] N. S. 19(61) (1946), 309–310 (Russian, with French summary). MR 0020248
- Steven R. Lay, Convex sets and their applications, Pure and Applied Mathematics, John Wiley & Sons, Inc., New York, 1982. MR 655598
- Frederick A. Valentine, Convex sets, McGraw-Hill Series in Higher Mathematics, McGraw-Hill Book Co., New York-Toronto-London, 1964. MR 0170264
Additional Information
- © Copyright 1990 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 108 (1990), 817-820
- MSC: Primary 52A35; Secondary 52A30
- DOI: https://doi.org/10.1090/S0002-9939-1990-0990413-5
- MathSciNet review: 990413