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.

We define the excess degree $\xi(P)$ of a $d$-polytope $P$ as $2f_1-df_0$, where $f_0$ and $f_1$ denote the number of vertices and edges, respectively. This parameter measures how much $P$ deviates from being simple. It turns out that the excess degree of a $d$-polytope does not take every natural number: the smallest possible values are $0$ and $d-2$, and the value $d-1$ only occurs when $d=3$ or 5. On the other hand, for fixed $d$, the number of values not taken by the excess degree is finite if $d$ is odd, and the number of even values not taken by the excess degree is finite if $d$ is even. The excess degree is then applied in three different settings. First, it is used to show that polytopes with small excess (i.e., $\xi(P)