The Lower Bound Theorem for $d$-Polytopes with $2{d}+1$ Vertices

Source abstract

The problem of calculating exact lower bounds for the number of k-faces of d-polytopes with n vertices, for each value of k, and characterizing the minimizers has recently been solved for n not exceeding 2 d. 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.