Projection inequalities for antichains

Abstract

A set $A\subseteq {\mathbb{R}}^n$ is called an antichain (resp. antichain) if it does not contain two distinct elements $\mathbf{x}=(x_1,\dots ,x_n)$ and $\mathbf{y}=(y_1,\dots ,y_n)$ satisfying $x_i\le y_i$ (resp. $x_i<y_i$) for all $i\in \{1,\dots ,n\}$. We show that the Hausdorff dimension of a weak antichain $A$ in the $n$-dimensional unit cube $[0,1{]}^n$ is at most $n-1$ and that the $(n-1)$-dimensional Hausdorff measure of $A$ is at most $n$, which are the best possible bounds. This result is derived as a corollary of the following {\it projection inequality}, which may be of independent interest: The $(n-1)$-dimensional Hausdorff measure of a (weak) antichain $A\subseteq [0,1{]}^n$ cannot exceed the sum of the $(n-1)$-dimensional Hausdorff measures of the $n$ orthogonal projections of $A$ onto the facets of the unit $n$-cube containing the origin. For the proof of this result we establish a discrete variant of the projection inequality applicable to weak antichains in ${\mathbb{Z}}^n$ and combine it with ideas from geometric measure theory.