  In 1967, Gr\"unbaum conjectured that any $d$-dimensional polytope with
$d+s\leq 2d$ vertices has at least \[\phi_k(d+s,d) = {d+1 \choose k+1 }+{d
\choose k+1 }-{d+1-s \choose k+1 } \] $k$-faces. We prove this conjecture and
also characterize the cases in which equality holds.
