On the Figiel-Lindenstrauss-Milman inequality

The Figiel-Lindenstrauss-Milman inequality is a fundamental inequality in the combinatorial theory of polytopes. It is classically obtained as a corollary of Milman’s version of Dvoretzky’s theorem. The goal of this paper is to provide a short and elementary proof of this inequality, derive more general versions of it, and discuss its tightness, where much is not known.