Separation by cylindrical surfaces

Lay, Steven R.

doi:10.1090/S0002-9939-1972-0310767-1

Separation by cylindrical surfaces
HTML articles powered by AMS MathViewer

by Steven R. Lay

Proc. Amer. Math. Soc. 36 (1972), 224-228

DOI: https://doi.org/10.1090/S0002-9939-1972-0310767-1

PDF | Request permission

Abstract:

C. Carathéodory has established that two compact sets P and Q in Euclidean n-space can be strictly separated by a hyperplane if each subset of $n + 1$ or fewer points of Q can be strictly separated from P by a hyperplane. In this paper it is shown that if each subset of k or fewer points of Q can be strictly separated from P by a hyperplane (where k is a fixed integer, $1 \leqq k \leqq n$), then there exists a cylinder of an appropriate sort containing P and disjoint from Q.

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
Olof Hanner and Hans Rådström, A generalization of a theorem of Fenchel, Proc. Amer. Math. Soc. 2 (1951), 589–593. MR 44142, DOI 10.1090/S0002-9939-1951-0044142-0

Über Mengen konvexer Körper mit gemeinschaftlichen Punkten

32

Alfred Horn, Some generalization of Helly’s theorem on convex sets, Bull. Amer. Math. Soc. 55 (1949), 923–929. MR 31759, DOI 10.1090/S0002-9904-1949-09309-6
Paul Kirchberger, Über Tchebychefsche Annäherungsmethoden, Math. Ann. 57 (1903), no. 4, 509–540 (German). MR 1511222, DOI 10.1007/BF01445182

Combinatorial separation theorems and convexity