Let $\Omega$ be a bounded open convex set of $R^{n}$, $u \in C^{0}(\bar{\Omega})$ a convex function, and $v$ a convex function whose graph is the upside down cone with vertex $(x_{0},u(x_{0}))$ and base $\Omega$.
I don't understand two properties of the function $v$. I think I lack some obious facts about $v$.
Can you show me, please, how to prove those two properties to help me to understand ?
1) Forall $x \in \Omega$, $u(x) \le v(x)$
2) Suppose forall $x \in \Omega, v(x) \ge v(x_{1}) + \langle p, x-x_{1} \rangle $ . If $x_{0} \ne x_{1}$ then $v(x_{1}) = v(x_{0}) + \langle p, x_{1}-x_{0} \rangle $
Thank you so much for your help !