I need help with the following problem, I have no idea how to proceed:
Let $u \colon \Omega \subseteq \mathbb{R}^N \to \mathbb{R}$ a continuous function, where $\Omega$ is open, connected and bounded. We define the set
$$\Gamma_u^+ = \{ y \in \Omega \colon (\exists p \in \mathbb{R}^N) \ (\forall x \in \Omega) \ u(x) \leq u(y) + \langle p,x-y \rangle\}$$ Prove that $f$ is concave iff $\Gamma_u^+ = \Omega$.
Any help will be appreciated.