Geodesic
On Semicontinuity of Convex-Valued Multifunctions and Cesari's Property (Q)
Journal of convex analysis, Tome 15 (2008) no. 4, pp. 803-818.

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\newcommand{\R}{\mathbb{R}} We investigate two types of semicontinuity for set-valued maps, Painlev\'{e}-Kuratowski semicontinuity and Cesari's property (Q). It is shown that, in the context of convex-valued maps, the concepts related to Cesari's property (Q) have better properties than the concepts in the sense of Painlev\'{e}-Kuratowski. In particular we give a characterization of Cesari's property (Q) in terms of upper semicontinuity of a family of scalar functions $\sigma_{f(\,\cdot\,)}(y^*) \colon X \to \overline\R$, where $\sigma_{f(x)} \colon Y^*\to \overline\R$ is the support function of the set $f(x)$. We compare both types of semicontinuity and show their coincidence in special cases.
Classification : 47H04,58C07
Mots-clés : Semi-continuity, set-valued maps, property (Q) 
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     author = {A. L\"ohne},
     title = {On {Semicontinuity} of {Convex-Valued} {Multifunctions} and {Cesari's} {Property} {(Q)}},
     journal = {Journal of convex analysis},
     pages = {803--818},
     publisher = {mathdoc},
     volume = {15},
     number = {4},
     year = {2008},
     url = {http://geodesic.mathdoc.fr/item/JCA_2008_15_4_JCA_2008_15_4_a7/}
}
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A. Löhne. On Semicontinuity of Convex-Valued Multifunctions and Cesari's Property (Q). Journal of convex analysis, Tome 15 (2008) no. 4, pp. 803-818. http://geodesic.mathdoc.fr/item/JCA_2008_15_4_JCA_2008_15_4_a7/