A Proof of Grünbaum's Lower Bound Conjecture for general polytopes

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

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