# Two combinatorial properties of a class of simplicial polytopes

## Abstract

Let f(P s d ) be the set of all f-vectors of simplicial d-polytopes. For P a simplicial 2 d-polytope let Σ(P) denote the boundary complex of P. We show that for each f ∈f(P s d ) there is a simplicial d-polytope P with f(P)=f such that the 11 02 simplicial diameter of Σ(P) is no more than f 0(P)-d+1 (one greater than the conjectured Hirsch bound) and that P admits a subdivision into a simplicial d-ball with no new vertices that satisfies the Hirsch property. Further, we demonstrate that the number of bistellar operations required to obtain Σ(P) from the boundary of a d-simplex is minimum over the class of all simplicial polytopes with the same f-vector. This polytope P will be the one constructed to prove the sufficiency of McMullen's conditions for f-vectors of simplicial polytopes. © 1984 Hebrew University.