Let $X$ be a separable Banach space, the associated dual space is denoted by $X^*$ and the usual duality between $X$ and $X^*$ by $\langle , \rangle$. For $C$ nonempty weakly compact convex subsets of $X$. we set $$ s(x^*, C) := \sup_ {x\in C} \langle x^* , x\rangle $$ to define for the set C its support function $s(., C)$.
The support function satisfies the following simple properties that are stated for easy references. $$ s(x^*, C + C')= s(x^*, C) + s(x^*, C'), \qquad x^*\in X^*,~ C, C' \in 2^X \setminus \{\emptyset\}. $$ $$ s(x^*,\alpha C)=\alpha s(x^*,C) , \qquad \alpha\geq 0,~x^*\in X^*,~ C \in 2^X \setminus \{\emptyset\}. $$ Let $x^*\in X^*$ and $h:\mathcal{P}_{wkc}\to \mathbb{R}$ a function defined by $C\mapsto s(x^*,C)$. With $$ \mathcal{P}_{wkc}(X)=\{C\in 2^X: C \text{ is nonempty convex weakly compact subset of }X\} $$ I found in an article: $ h $ is an affine function. But I don't understand why.
Is Minkowski's sum considered in $\mathcal{P}_{wkc}$ ?