Suppose that$\displaystyle $S\subset X$$ is such that $\displaystyle $\forall x \in S$, $\forall \zeta \in N_S^P(x)$$, we have $\displaystyle $$\langle \zeta, x'-x\rangle \leq 0 \ \ \forall x'\in S$$$. Prove that S is convex. Here X is a Hilbert space and $\displaystyle $N_S^P(x)$$ means the proximal normal cone.

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