Abstract
Suppose F is a convex functional defined on a normed linear space X, M ∋ X, and\(M_F = \{ x \in M:F(x) = \mathop {\inf }\limits_{\upsilon \in M} F(\upsilon )\}\). The mapping which assigns the set mf to each convex set M in X (mf ≠ ø) is considered. It is proved that this mapping is uniformly continuous if and only if the functional F is uniformly convex.
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Translated from Matematicheskie Zametki, Vol. 19, No. 4, pp. 501–512, April, 1976.
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Berdyshev, V.I. Continuous dependence of an element realizing the minimum of a convex functional on the set of admissible elements. Mathematical Notes of the Academy of Sciences of the USSR 19, 307–313 (1976). https://doi.org/10.1007/BF01156788
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DOI: https://doi.org/10.1007/BF01156788