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Remarks on multifunctions with convex graph. (English) Zbl 0648.46010
The following result is proved:
Theorem 1. Let X, Y be two real topological vector spaces; X a non-empty open convex subset of E; F a lower semicontinuous multifunction from X into Y, with non-empty closed values and convex graph (or even midpoint convex graph). Then, the graph of F is closed in $$X\times Y.$$
Applying Theorem 1, we also get the following
Theorem 2. Let X, Y be two real Banach spaces and let F be a multifunction from X onto Y, with non-empty values and convex graph. Then, the following assertions are equivalent:
(a) The multifunction F is lower semicontinuous and closed-valued.
(b) The graph of the multifunction F is closed in $$X\times Y.$$
(c) The multifunction F is open and, for each $$y\in Y$$, the set $$F^ - (y)$$ is closed in X.
Reviewer: B.Ricceri

MSC:
 46A55 Convex sets in topological linear spaces; Choquet theory 54C60 Set-valued maps in general topology
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References:
 [1] B. Ricceri, On multifunctions with convex graph. Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur.77, 64-70 (1984). · Zbl 0611.54011 [2] S. M. Robinson, Regularity and stability for convex multivalued functions. Math. Oper. Res.1, 130-143 (1976). · Zbl 0418.52005 · doi:10.1287/moor.1.2.130
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