On the diameter of convex polytopes

In this communication we prove a theorem on the diameter of convex polytopes, which generalizes a theorem of Naddef [4]. For a recent survey on diameters of polytopes and the related d-step conjecture, the reader can consult PI. For a polytope P in R”, we denote by dim(P) its affine dimension, by vert(P) its vertex set, by d,(u, V) the distance between two vertices on its l-skeleton, and by 6(P) its diameter. The symmetrization of P is denoted by P P = {x -y: x, y E P}. The linear span of a subset S of R” is denoted by lin(S), and the orthogonal complement of a subspace L by LI. For a linear functional w on R”, denote by k(P, w) the number of distinct values attained by the vertices of P under w, minus one, i.e. k(P, w) = I{( w, v): v E vert(P)}I 1, and by F_(P, w) and F+(p, w) denote the faces of P containing those vertices attaining the minimum and maximum values under w, respectively.