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We consider simplicial polytopes, and more general simplicial complexes, without missing faces above a fixed dimension. Sharp analogues of McMullen's generalized lower bounds, and of Barnette's lower bounds, are conjectured for these families of complexes. Some partial results on these conjectures are presented.arXiv:0810.5487v2 fatcat:xg27m6m4nfawpl62czog7tbis4