Lemma. Let P be a polyhedron (convex, but not necessarily compact), p a point on a facet Φ of P, and f a nonzero affine-linear function vanishing at p. If Φ contains a neighborhood of p in f−1(0),then p is an interior point of Φ, the equation of Φ is f=0, and the inequality determining Φ is either f ≥ 0 or f ≤ 0.

This lemma from the following paper (P25 Lemma 1):