Convex hull of finite union

Question

Suppsose $A\subset\mathbb{R}^n$ and let co$(A)$ be its convex hull. Then does the following hold

for $A_1,\cdots,A_k$: $\text{co}(A_1\cup\cdots\cup A_k)=\text{co}(\text{co}(A_1)\cup\cdots\cup\text{co}(A_k))$?

Answer 1

Yes, the "$\subseteq$" is trivial since $A_j\subseteq co(A_j)$.

For "$\supseteq$" note that $co(A_1\cup\cdots \cup A_k)$ contains $co(A_j)$ for any $j=1,\cdots,k$. Thus $co(A_1\cup\cdots \cup A_k)$ is a convex set containing $co(A_1)\cup \cdots \cup co(A_k)$, hence it contains $co(co(A_1)\cup \cdots \cup co(A_k))$.