A lower bound theorem for $d$-polytopes with $2d+1$ vertices

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 characterising the minimisers, has recently been solved for $n\le2d$. We establish the corresponding result for $n=2d+1$; the nature of the lower bounds and the minimising polytopes are quite different in this case. As a byproduct, we also characterise all $d$-polytopes with $d+3$ vertices, and only one or two edges more than the minimum.