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

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