Abstract
Replacing “convex” by “strongly convex” we show that Helly's famous intersection theorem holds on every Riemannian n-manifold in the following form: The intersection of k relatively compact, strongly convex subsets of M (k≧n+i≧2) is nonvoid as soon as any n+i of these sets have a nonvoid intersection, where i=2 if M is homeomorphic to the standard n-sphere and i=1 otherwise.
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DANZER, L., GRÜNBAUM, B., KLEE, V.: Helly's theorem and its relatives. Proc. of Symposia in Pure Mathematics, Amer. Math. Soc.,7, Convexity 100–181 (1963)
DEBRUNNER, H.E.: Helly type theorems derived from basic singular homology. Amer. math. monthly77, 375–380 (1970)
HELLY, E.: über Mengen konvexer Körper mit gemeinschaftlichen Punkten. Jber. Deutsch. Math. Verein32, 175–176 (1923)
HELLY, E.: über Systeme von abgeschlossenen Mengen mit gemeinschaftlichen Punkten. Monatsh. Math. und Physik37, 281–302 (1930)
KARCHER, H.: Schnittort und konvexe Mengen in vollständigen Riemannschen Mannigfaltigkeiten. Math. Ann.177, 105–121 (1968)
KOBAYASHI, S., NOMIZU, K.: Foundations of differential geometry II. New York-London-Sydney: Interscience 1969
ROSEN, R.: A weak form of the star conjecture for manifolds. Abstract 570-28, Notices Amer. Math. Soc.7, 380 (1960)
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Kleinjohann, N. Remark on the Helly number for strongly convex sets on Riemannian manifolds. Manuscripta Math 34, 27–29 (1981). https://doi.org/10.1007/BF01168707
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DOI: https://doi.org/10.1007/BF01168707