Abstract
We investigate the class of generalized convex sets on Grassmann manifolds, which includes known generalizations of convex sets for Euclidean spaces. We extend duality theorems (of polarity type) to a broad class of subsets of the Euclidean space. We establish that the invariance of a mapping on generalized convex sets is equivalent to its affinity.
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REFERENCES
Yu. B. Zelinskii, Many-Valued Mappings in Analysis [in Russian], Naukova Dumka, Kiev (1993).
V. A. Rokhlin and D. B. Fuks, Beginner's Course in Topology [in Russian], Nauka, Moscow (1977).
F. E. Goodmen, “When is a set of lines in space convex?,” Notic. Amer. Math. Soc. 45, No. 2, 222–232 (1998).
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Zelinskii, Y.B., Momot, I.V. On (n, m)-Convex Sets. Ukrainian Mathematical Journal 53, 482–487 (2001). https://doi.org/10.1023/A:1012352607365
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DOI: https://doi.org/10.1023/A:1012352607365