On Semicontinuity of Convex-Valued Multifunctions and Cesari's Property (Q)

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.