THE LOWER BOUND THEOREM FOR d -POLYTOPES WITH 2d + 1 VERTICES

The problem of calculating exact lower bounds for the number of k-faces of dpolytopes with n vertices, for each value of k, and characterizing the minimizers has recently been solved for n not exceeding 2d. We establish the corresponding result for n = 2d+ 1; the nature of the lower bounds and the minimizing polytopes are quite different in this case. As a byproduct, we also characterize all d-polytopes with d + 3 vertices and only one or two edges more than the minimum.