Abstract
By using the classical Helly theorem, one cannot obtain information about a family of convex compact sets in the n-dimensional Euclidean space if it is known that only subfamilies consisting of k elements, 0 < k ≤ n, have nonempty intersections. We modify the Helly theorem to fix this issue and investigate the behavior of generalized convex families.
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Yu. B. Zelinskii, Multivalued Mappings in Analysis [in Russian], Naukova Dumka, Kiev (1993).
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Zelinskii, Y.B. Helly Theorem and Related Results. Ukrainian Mathematical Journal 54, 149–153 (2002). https://doi.org/10.1023/A:1019753822066
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DOI: https://doi.org/10.1023/A:1019753822066