For convex functions $g_1, \dots, g_m: (\mathcal{D} \subseteq \mathbb{R}^d) \to \mathbb{R}$, define $g(\boldsymbol{x}):= \max_{1 \leq i \leq m} g_i (\boldsymbol{x})$. Show that $$\partial g(\boldsymbol{x}) = \mathrm{conv} \Big( \cup_{i: g_i(\boldsymbol{x}) = g(\boldsymbol{x})} \partial g_i (\boldsymbol{x}) \Big).$$
It's not hard to prove the right is a subset of the left one. But I have no idea for the opposite direction