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

We investigate two types of semicontinuity for set-valued maps, Painlevé-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 Painlevé-Kuratowski. In particular we give a characterization of Cesari's property (Q) in terms of upper semicontinuity of a family of scalar functions

, where

\sigma_{f(x)} \colon Y^*\to \overline{\mathbb{R}} σ f ( x )  ⁣ : Y ∗ → R ‾

is the support function of the set

f(x) f ( x )

. We compare both types of semicontinuity and show their coincidence in special cases.