Let's assume that we have a simplex $G = \{x\in R^d|\sum_{i=1}^d x_i=1, x_i\ge 0 , i = 1, 2, .., d\}$ and a polyhedral convex subset $H \subseteq G$.

Is it possible to prove that for any extremal point $x^*\in H$, the tangential surface of $x^*$ is a supporting hyperplane of $H$?

In a simple 2D case, it is somewhat trivial. However, for general high dimensional cases, it wasn't..

Thank you in advance. :)