McMullen's conditions and some lower bounds for general convex polytopes

We give a lower bound for the number of vertices of a generald-dimensional polytope with a given numberm ofi-faces for eachi = 0,..., ⌎d/2⌏ − 1. The tightness of those bounds is proved using McMullen's conditions. Form greater than a small constant, those lower bounds are attained by simpliciali-neighbourly polytopes.