grandes-ecoles 2025 Q18

grandes-ecoles · France · x-ens-maths-d__mp Proof Proof of Set Membership, Containment, or Structural Property

We will admit that for every non-empty closed convex set $C \subset \mathbb{R}^n$ and every $x \in \mathbb{R}^n \backslash C$, there exists a unique $y \in C$ such that $\langle x - y, z - y \rangle \leq 0$ for all $z \in C$.
Let $V \subset \mathbb{R}^n$ be a non-empty finite set. Prove that every vertex of $\operatorname{Conv}(V)$ belongs to $V$.

We will admit that for every non-empty closed convex set $C \subset \mathbb{R}^n$ and every $x \in \mathbb{R}^n \backslash C$, there exists a unique $y \in C$ such that $\langle x - y, z - y \rangle \leq 0$ for all $z \in C$.

Let $V \subset \mathbb{R}^n$ be a non-empty finite set. Prove that every vertex of $\operatorname{Conv}(V)$ belongs to $V$.

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