grandes-ecoles 2022 Q6

grandes-ecoles · France · x-ens-maths__psi_cpge Proof Existence Proof

Let $C$ and $D$ be two non-empty convex subsets of $\mathbb{R}^d$ such that $C$ is closed and bounded, $D$ is closed, and $C \cap D = \emptyset$. Show that there exist $p \in \mathbb{R}^d$ and $\varepsilon > 0$ such that $$p \cdot x \leqslant p \cdot y - \varepsilon, \forall (x, y) \in C \times D.$$ (we say that $C$ and $D$ can be strictly separated).

Let $C$ and $D$ be two non-empty convex subsets of $\mathbb{R}^d$ such that $C$ is closed and bounded, $D$ is closed, and $C \cap D = \emptyset$. Show that there exist $p \in \mathbb{R}^d$ and $\varepsilon > 0$ such that
$$p \cdot x \leqslant p \cdot y - \varepsilon, \forall (x, y) \in C \times D.$$
(we say that $C$ and $D$ can be strictly separated).

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