Minimization over a convex subset characterized by subgradient

Question

Problem: Let $f: \mathbb{R}^{n} \to \mathbb{R}$ be a proper, convex function. $C$ is a closed subset of $\mathbb{R}^n$ and $x^*$ is a minimizer of $f$ over C. Prove that there exists a subgradient $g\in \partial f(x^*)$ such that \begin{equation} g^T(x-x^*)\geq0, \forall x\in C \end{equation} any help is appreciated!

Answer 1

A subgradient is a directional derivative. If we look at the directional derivative at $x^*$ then we have $$0\overset{(*)}{\leq}\lim_{\alpha\to0^+}\frac{f(x^*+\alpha(x-x^*))-f(x^*)}{\alpha}=\partial f(x^*)^T(x-x^*)=g^T(x-x^*)$$ where $(*)$ come from the fact that $x^*$ is a global minimum of $f$.