From the introductory chapter of Itamar Pitowsky's Quantum Probability — Quantum Logic:
Under the second description, a vector is an element of the polytope if and only if its coordinates satisfy a set of linear inequalities which represent the supporting hyperplanes of the polytope. The existence of such a dual descripton for every polytope is known as the Weyl-Minkowski theorem.
What exactly is Weyl-Minkowski theorem? I looked online but could not understand much as it has something to do with duality in linear programming which I have only a vague idea about.
Like Math1000 wrote, basically there are two ways to define a polytope: by its faces or by its vertices. Weyl-Minkowski theorem says these two descriptions always exist (i.e., are equivalent).