Abstract. Consider a lower semicontinuous nonempty closed convex set-valued map $Q$ from a compact Hausdorff topological space ${\cal T}$ to ${\msbm R}^r$. To $Q$, there correspond a subset ${\msbm Q}$ of $C({\cal T},{\msbm R}^r)$ and a functional $q$ on ${\cal T}\times{\msbm R}^r$. Expressions for the tangent cone of ${\msbm Q}$ is given in terms of the corresponding concepts for $Q(t)$. The image space of each of the maps $Q \mapstochar\rightarrow {\msbm Q}$ and $Q \mapstochar\rightarrow q$ is completely described for this case and for the case when $Q(t)$ is open or has a nonempty interior for all $t$ in ${\cal T}$.

AMS Subject Classification (1991): 54C60, 54C65; 47H04, 58C06

Keyword(s): Set-valued maps, lower semicontinuous, C, C({\cal T}, closed and open-convex sets in, {\msbm R}^r), support functional, tangent and normal cones

Received March 10, 1998 and in revised form December 17, 1998. (Registered under 2689/2009.)