I am trying to prove that $\newcommand{\co}{\overline{\mathrm{co} }} \co(A)-\co(B) \subseteq \co(A-B)$ in a linear topological space (topological vector space) $X$.
Here $\co(A)$ is the closed convex hull of $A$, the closure of the convex hull of $A$ in $X$.
Here is some argument which is quite vague for me: Since $x \to a-x$ is an affine homeomorphism for all $a\in A$, we get $a-\co(B)=\co(a-B) \subseteq \co(A-B)$. Then for arbitrary fixed $b \in \co(B)$, we have $a-b \in \co(A-B)$ for all $a \in A$. Hence $\co(A)-b=\co(A-b)\subseteq \co(A-B)$ for all $b \in \co(B)$. At this point the claim is that the proof is complete.
Any other more direct proof or explanation of this one in detail would help.