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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 andalso characterize the cases in which equality holds.

A Proof of Gr\"unbaum's Lower Bound Conjecture for general polytopes