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 num ber upsilon of vertices, the maximum number of inner diagonals is [GRAPHICS] when d greater than or equal to 4 it is attained if and only if P is a stac ked 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 greater than or equal to 6, only by) certain simple polytopes. (c ) When d = 3, the maximum in (b) is determined for all f; when f greater th an or equal to 14 it is 2f(2) - 21f + 64 and the unique associated p-vector is 5(12)6(f/-12). (d) Among all simple 3-polytopes with f Facets, the mini mum number of inner diagonals is f(2)-9f+20; when f greater than or equal t o 9 the unique associated p-vector is 3(2)4(f-4)(f-1)(2) and the unique ass ociated combinatorial type is that of the wedge over an (f-1)-gon. (C) 1999 Academic Press.