Cutting a Polytope

摘要

We show that given two vertices of a polytope one cannot in general find a hyperplane containing the vertices that has two or more facets of the polytope in one closed half-space. Our result refutes a long-standing conjecture. We prove the result by constructing a 4-dimensional polytope that provides the counterexample. Also, we show that such a cutting hyperplane can be found for each pair of vertices, if the polytope is either simplicial or 3- dimensional.