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Register a new userA Proof of Grunbaums Lower Bound Conjecture for general polytopes
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Lei Xue
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In 1967, Grunbaum conjectured that any $d$-dimensional polytope with $d+sleq 2d$ 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.
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A remarkable and important property of face numbers of simplicial polytopes is the generalized lower bound inequality, which says that the $h$-numbers of any simplicial polytope are unimodal. Recently, for balanced simplicial $d$-polytopes, that is simplicial $d$-polytopes whose underlying graphs are $d$-colorable, Klee and Novik proposed a balanced analogue of this inequality, that is stronger than just unimodality. The aim of this article is to prove this conjecture of Klee and Novik. For this, we also show a Lefschetz property for rank-selected subcomplexes of balanced simplicial polytopes and thereby obtain new inequalities for their $h$-numbers.
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