A d-PSEUDOMANIFOLD WITH fo VERTICES HAS AT LEAST dfo-(d-1)(d+2) d-SIMPLICES

Barnette was the first o prove that if fk is the number of k-faces of a simple (d+l)-polytope P then (*) fo • dfd ' (d-1)(d+2). He later extended (*) to a graph-theoretic setting and was thereby enabled to prove the dual inequality for triangulated d-manifolds. Here his methods are used to provide a different graph-theoretic extension of (*) and thus extend the dual inequality to simplicial d-pseudomanifolds. Introduction. A d-polytope is a d-dimensional set (in a real vector space) that is the convex hull of a finite set, and it is simple if each of its vertices is incident to precisely d edges. The inequality (*) becomes an equality when d C ( 1,2), as follows readily from Euler's theorem, and also when P is obtained from a d-simplex by successive truncations of vertices. Though (*) was first stated for d = 3 by BrCickne