Show that if $f : \mathbb{R}^n \rightarrow \mathbb{R}$ is differentiable at $x^{\star} \in K$ and admits a local minimum on $K$ at $x^{\star}$, then
$$\forall h \in \mathcal{A}_K(x^{\star}), \quad \langle \nabla f(x^{\star}), h \rangle \geqslant 0.$$
What does this result express in the particular case where $x^{\star}$ is in the interior of $K$?