Inner Diagonals of Convex Polytopes

An inner diagonal of a polytope P is a segment that joins two vertices of P and that lies, except for its ends, in P's relative interior. The paper's main results are as follows: (a) Among all d-polytopes P having a given number v of vertices, the maximum number of inner diagonals is [[formula]]−dv+[[formula]]; when d⩾4 it is attained if and only if P is a stacked polytope. (b) Among all d-polytopes having a given number f of facets, the maximum number of inner diagonals is attained by (and, at least when d=3 and f⩾6, only by) certain simple polytopes. (c) When d=3, the maximum in (b) is determined for all f; when f⩾14 it is 2f2−21f+64 and the unique associated p-vector is 5126f−12. (d) Among all simple 3-polytopes with f facets, the minimum number of inner diagonals is f2−9f+20; when f⩾9 the unique associated p-vector is 324f−4 (f−1)2 and the unique associated combinatorial type is that of the wedge over an (f−1)-gon.