Abstract
For a familyA of closed bounded convex subsets of a Banach space, sufficient conditions are given for the existence of a closed hyperplane which supports each member ofA.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Bishop, E. and Phelps, R. R.: The support functionals of a convex set,Proc. Symp. Pure Math. Amer. Math. Soc. 7 (Convexity) (1963), 27–35.
Bisztriczky, T.: On separated families of convex bodies,Arch. Math. 54 (1990), 193–199.
Carathéodory, C.: Über den Variabilitätsbereich der Koeffizienten von Potenzreihen, der gegebene Werte night annehmen,Math. Ann. 64 (1907), 95–115
Collier, J. B. and Edelstein, M.: On strongly exposed points and Fréchet differentiability,Israel J. Math. 17 (1974), 66–68.
Collier, J. B. and Edelstein, M.: Common supports for pairs of sets,Israel J. Math. 19 (1974), 237–240.
Dawson, R. and Edelstein, M.: Families of bodies with definite common supports,Geom. Dedicata 33 (1990), 195–204.
Dawson, R.: Helly-type theorems for bodies in the plane with common supports,Geom. Dedicata 45 (1993), 289–299.
Edelstein, M. and Thompson, A. C.: Some results on nearest points and support properties of convex sets inc 0,Pacific J. Math. 40 (1972), 553–560.
Helly, E.: Über Mengen konvexer Körper mit gemeinschaftlichen Punkten,Jahrb. Deut. Math. Verein 32 (1923), 175–176.
Phelps, R. R.: Dentability and extreme points in Banach spaces,J. Func. Anal. 16 (1974), 78–90.
Singer, I.: Some extensions of a dual of the Hahn-Banach theorem, with applications to separation and Helly-type theorems.Austral. Math. Soc. Bull. 15 (1976), 277–291.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Dawson, R. Common supports of families of sets. Geom Dedicata 51, 1–13 (1994). https://doi.org/10.1007/BF01264097
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01264097