Flattening Antichains with Respect to the Volume

Abstract

We say that an antichain $\cal A$ in the boolean lattice $B_n$ is flat if there exists an integer $k\geq 0$ such that every set in $\cal A$ has cardinality either $k$ or $k+1$. Define the volume of $\cal A$ to be $\sum_{A\in{\cal A}}|A|$. We prove that for every antichain $\cal A$ in $B_n$ there exist an antichain which is flat and has the same volume as $\cal A$.