From Membership to Separation, a Simple Construction

In [3], Martin Grötschel, László Lovász and Alexander Schrijver use a construction of Dmitrii Yudin et Arkadiĭ Nemirovskiĭ to polynomially separate a point x from a centered bounded convex K using a membership oracle. In this note, we present a natural and simple construction which solve the same problem but for the simpler case of polyhedra. Namely, given a well defined polyhedron P with a non-empty interior, a point $x \notin P$ and a point $a \in \operatorname{int} {\left( P \right)}$, using a polynomial number of calls of the membership oracle, we find a facet of P whose supporting hyperplane separates x from P.