Currently I am studying Behrend's $M$-structure and Banach-Stone Theorem. He introduced the following notation.
Notation: Consider a Banach space $X.$ Fix $x\in X$ and $r\geq 0.$ Consider the set $$K(x,r)=\{(x^*,x^*(x)+r)\in X^*\times \mathbb{R}:x^*\in B_{X^*}\}.$$
It is not hard to show that $K(x,r)$ is a compact convex set.
However, the author quoted the following at page $45.$
convex hull of $\bigcup_{i=1}^nK(x_i,r_i)$ is compact because convex hull of finite unions of compact convex sets is compact.
I fail to prove the above statement. Any hint is appreciated.
The convex hull of a set is the set of convex combinations of elements of that set. I claim that the convex hull of an union of (non-empty) convex sets $C_1\cup \cdots \cup C_n$ is the set of convex combinations in the form $t_1a_1+\cdots +t_na_n$, where $a_i\in C_i$ for all $i$. In fact, consider $x=\sum_{i,j} t_{i,j} a_{i,j}$, where $a_{i,j}\in C_i$ and $t_{i,j}>0$ for all $i,j$, plus $\sum_{i,j} t_{i,j}=1$. Then, call $a_i=\sum_{j} \frac{t_{i,j}}{\sum_{j} t_{i,j}}a_{i,j}$. This is a convex combination of elements of $C_i$, and thus $a_i\in C_i$. Moreover, $$x=\sum_i \left(\sum_{j} t_{i,j}\right)a_i,$$ which is indeed a convex combination with at most one element from each $C_i$.
Now, the same argument you used for $\Bbb R^d$ - be it sequential compactness, or $\operatorname{co}\left(\bigcup_{i=1}^n C_i\right)$ being image of a closed subset of $[0,1]^n\times \prod_{i=1}^n C_i$ by a continuous function - carries on here.