One functional-analytical idea by Alexandrov in convex geometry.
Kutateladze, S.S. (2002)
Vladikavkazskiĭ Matematicheskiĭ Zhurnal
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Kutateladze, S.S. (2002)
Vladikavkazskiĭ Matematicheskiĭ Zhurnal
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V. I. Oliker (1982)
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze
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Annales de l'institut Fourier
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This paper gives necessary and sufficient conditions for the closure of a face in a compact convex set to be again a face. As applications of these results, several theorems scattered in the literature are proved in an economical and uniform manner.
Thomas, E. G. F.
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Firey, W.J. (1966)
Portugaliae mathematica
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The paper begins with a self-contained and short development of Bárány’s theorems of Carathéodory and Helly type in finite-dimensional spaces together with some new variants. In the second half the possible generalizations of these results to arbitrary Banach spaces are investigated. The Carathéodory-Bárány theorem has a counterpart in arbitrary dimensions under suitable uniform compactness or uniform boundedness conditions. The proper generalization of the Helly-Bárány theorem reads...