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On Lipschitz midconvex multifunctions. (English) Zbl 1103.26011
It is proved that a normed space-valued midconvex multifunction with non-empty closed and bounded values, weakly upper bounded on a suitable ball of a normed space $$X$$, is locally Lipschitzian. The proof utilizes some technical results, which are demonstrated in the first part of the paper. Moreover, as a corollary, a sufficient condition for Lipschitzianity of $$F$$ on the whole of $$X$$ is given.

##### MSC:
 26A51 Convexity of real functions in one variable, generalizations
##### Keywords:
multifunction; convexity; Lipschitz condition
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##### References:
 [1] NIKODEM K.: K-convex and K-concave set-valued functions. Zeszyty Nauk. Politech. Łódz. Mat. 559 (1989), 1-75. [2] PSHENICHNYI B.: Convex multivalued mappings and their conjugates. Cybernetics 10 (1974), 453-464. [3] ROBERTS A. W.-VARBERG D. E.: Convex functions. New York-London, Academic Press, 1973. · Zbl 0271.26009
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