I'm looking to prove a intuitive enough propriety of the subdifferentials of a convex function : let $f : E \to \mathbb{R}$ be a convex function, $n \in \mathbb{N}$ and let $p_1,p_2,\dots,p_n \in E^n$. Let now $p \in \operatorname{Conv}(p_1,p_2,\dots,p_n)$, where $\operatorname{Conv}$ is the convex hull. My intent is then to prove that $$\partial f(p) \subset \operatorname{Conv}(\partial f(p_1), \partial f(p_2), \dots, \partial f(p_n)).$$
If $E = \mathbb{R}$, the monotonicity of $\partial f$ kills the problem, but I have not been able to replicate the proof for higher dimensions. The monotonicity of $\partial f$ seems to be a key element of the proof, but I haven't been able to use its full potential. Even if I don't think so, this result might require $p$ to be in the interior of $\operatorname{Conv}(p_1,p_2,\dots,p_n)$.
If it simplifies the problem, I am particularly interested in the case where $E = \mathbb{R}^d$ and $n = d+1$. In my case of study, all subsets of $\{p_1,\dots,p_{d+1}\}$ of cardinal $d$ defines a base of $\mathbb{R}^d$, and $f$ is moreover differentiable, but I don't think those assumptions are necessary.
Any help would be welcome, thanks in advance !
Ivan