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I was reading about convex hulls on Wikipedia (Convex hull) and I read :

$ Conv(A \cup B)= Conv(Conv(A) \cup Conv (B))$ where $A$ and $B$ are nonempty sets.

I can see intuitively that this equality is true, but I do not know how to write it formally down.

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1 Answers1

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For simplicity, let us denote $T$ as the operation of taking the convex hull of a set.

We observe following basic principles.

  1. $A \subset T(A)$.

  2. If $A \supset B$, then $T(A) \supset T(B)$.

  3. $T(T(A))=T(A)$.

Now we prove the proposition.

From 2, $T(A \cup B) \supset T(A)$ and also $T(A \cup B ) \supset T(B)$, so $T(A \cup B) \supset T(A) \cup T(B)$. Hence $T(A \cup B) \supset T(T(A) \cup T(B))$.

On the other hand, we have $ A \cup B \subset T(A) \cup T(B)$. Therefore, $T(A \cup B) \subset T(T(A) \cup T(B))$.

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