Let $V \subset \mathbb{R}^n$ be a non-empty finite set. We assume that $\operatorname{Conv}(V)$ is not contained in a hyperplane of $\mathbb{R}^n$ and contains 0 in its interior, and that the set $Q = \{\ell \in \mathbb{R}^n : \langle \ell, x \rangle \leq 1\ \forall x \in V\}$ is a polytope of $\mathbb{R}^n$. Deduce that $\operatorname{Conv}(V)$ is a polytope.