## vertices of Cartesian product of polytopes

If a convex polytope is in a Cartesian product form as $P = \prod_{k=1}^{K}P_k$, then the vertices of $P$, $V$ are obtained by composing all the possible combinations, i.e., $V = (V_1,\ldots,V_K)$, where $V_k$ are the vertices of $P_k$.

I believe this is true and can prove it for a polytope with a special structure. Can someone refer me to some textbooks or papers stating the above as a theorem or a lemma? (I searched, but did not find something useful) thx a lot!

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I bet you can prove it yourself by using the fact that the vertices are exactly the points that are not convex combinations of a pair of two other points in the polytope. – Yoav Kallus Jun 12 at 5:15