Separating two compact sets by a parallelotope
HTML articles powered by AMS MathViewer
- by Steven R. Lay PDF
- Proc. Amer. Math. Soc. 79 (1980), 279-284 Request permission
Abstract:
Let P and Q be compact subsets of Euclidean n-space. Paul Kirchberger has established that there exists a closed half-space M such that $P \subset M$ and $Q \cap M = \emptyset$ iff for each set S consisting of $n + 2$ or fewer points of $P \cup Q$, there exists a closed half-space ${M_S}$ such that $(P \cap S) \subset {M_S}$ and $(Q \cap S) \cap {M_S} = \emptyset$. In this paper the problem of replacing the half-spaces by parallelotopes is considered, and the critical number of points in $P \cup Q$ is shown to be $n + 1$. Applications of this are drawn to systems of linear inequalities and to Carathéodory’s theorem.References
- C. Carathéodory, Über den Variabilitätsbereich der Koeffizienten von Potenzreihen, die gegebene Werte nicht annehmen, Math. Ann. 64 (1907), no. 1, 95–115 (German). MR 1511425, DOI 10.1007/BF01449883
- Paul Kirchberger, Über Tchebychefsche Annäherungsmethoden, Math. Ann. 57 (1903), no. 4, 509–540 (German). MR 1511222, DOI 10.1007/BF01445182
- S. R. Lay, On separation by spherical surfaces, Amer. Math. Monthly 78 (1971), 1112–1113. MR 300201, DOI 10.2307/2316320
- Steven R. Lay, Separation by cylindrical surfaces, Proc. Amer. Math. Soc. 36 (1972), 224–228. MR 310767, DOI 10.1090/S0002-9939-1972-0310767-1
- 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 1980 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 79 (1980), 279-284
- MSC: Primary 52A35
- DOI: https://doi.org/10.1090/S0002-9939-1980-0565354-1
- MathSciNet review: 565354