It is proved that equality in the Generalized Simplicial Lower Bound Conjecture can always be obtained by k-stacked polytopes.
Let P be a simplicial convex d-polytope with f~ faces of dimension i. The vector f(P) = (f0 ..... fa-1) is called the f-vector of P. The complete characterization of all f-vectors, known as McMullen's g-conjecture [3], has been obtained by Billera and Lee in and by Stanley in . Billera and Lee proved the sufficiency and Stanley the necessity of McMuUen's conditions for a vector in Z a to be the f-vector of some simplicial d-polytope. These conditions are formulated in terms of the h-vector of a polytope rather than in terms of the f-vector.
The vector h(P) = (h0, hi ..... ha) is called the h-vector of P, where hi--,=0-t (d d ~ I.)(-1)i-if/-1 Oc-1 =: 1).
Then the g-conjecture (or rather the g-Theorem) may be formulated as follows:
A vector h = (ho ..... ha) in 7/a+l is the h-vector of some simplicial d-polytope if and only if the following conditions hold: (i) ~=ha-i, O<~i~n:=[f2d], (ii) h/.~>~_l, l<~i<~n, (iii) h0 = 1 and ~+1 -hi ~< (~ -~-1) ~i>, 1 ~< i ~< n -1. (For the definition of the functional x (i> see [1], [3] or [8].) The inequality (ii) together with the following condition for equality is known as the "Generalized Simplicial Lower Bound Conjecture" first formulated by McMullen and Walkup [4]: