Let $S = \{x_0, \dots, x_n\} \subseteq (\mathbb{R}^+)^{d}$ be the set of vertices of a convex $d$-dimensional convex polytope ($d \geq 2$).
I am interested in finding a set of linear inequalities such that a point $x$ is inside the polytope iff $x$ satisfies all the inequalities. I have seen discussions about the converse problem (find vertices from the set of constraints), but not this one. The context is to use those inequalities as constraints in a linear program.