  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.
