Remarks on multifunctions with convex graph.

*(English)*Zbl 0648.46010The 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.

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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