Let $(A\_i: i \in I)$ be a family of sets in a topological vector space such that for all $i,j \in I$ there exists $k \in I$ for which $A\_i \cap A\_j=A\_k$. Denote by $\overline{co}(X)$ the closure of the convex hull of $X$. Is it true that
$$
\bigcap\_{i \in I}\overline{co}(A\_i)=\overline{co}\left(\bigcap\_{i \in I}\overline{A\_i}\right)\,\,\,?
$$

One inclusion: Since $\overline{A\_i} \subseteq \overline{co}(A\_i)$ for all $i$ then $\bigcap\_i \overline{A\_i} \subseteq \bigcap\_i \overline{co}(A\_i)$. Therefore
$$
\overline{co}\left(\bigcap\_i \overline{A\_i}\right) \subseteq \overline{co}\left(\bigcap\_i \overline{co}(A\_i)\right)=\bigcap\_i \overline{co}(A\_i).
$$

What about the other?

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**Comment 1.** The hypothesis on the family $\{A\_i:i \in I\}$ is necessary, see Kavi's answer.

**Comment 2.** The claim is easily seen to be verified if $\{A\_i: i \in I\}$ has a minimum, which holds, e.g., if $I$ is finite.

**Comment 3.** The claim $\bigcap\_{i}\overline{co}(A\_i)=\overline{co}\left(\bigcap\_{i}A\_i\right)$ (i.e., replacing $\overline{A\_i}$ with $A\_i$ in the right side) is false. As a counterexample, let $A\_i=(0,2^{-i})\cup\{1\}$ for all integers $i\ge 1$. Then the left side is $[0,1]$ and the right side is $\{1\}$.