A Positive Answer to Bárány's Question on Face Numbers of Polytopes.

Despite a full characterization of the face vectors of simple and simplicial polytopes, the face numbers of general polytopes are poorly understood. Around 1997, Bárány asked whether for all convex d-polytopes P and all 0 ≤ k ≤ d - 1 , f k (P) ≥ min { f 0 (P) , f d - 1 (P) } . We answer Bárány's question in the affirmative and prove a stronger statement: for all convex d-polytopes P and all 0 ≤ k ≤ d - 1 , f k (P) f 0 (P) ≥ 1 2 [ ⌈ d 2 ⌉ k + ⌊ d 2 ⌋ k ] , f k (P) f d - 1 (P) ≥ 1 2 [ ⌈ d 2 ⌉ d - k - 1 + ⌊ d 2 ⌋ d - k - 1 ]. In the former, equality holds precisely when k = 0 or when k = 1 and P is simple. In the latter, equality holds precisely when k = d - 1 or when k = d - 2 and P is simplicial. [ABSTRACT FROM AUTHOR]