# Projection inequalities for antichains

## TL;DR

This paper establishes optimal bounds on the Hausdorff dimension and measure of antichains in the unit cube, deriving a novel projection inequality that relates the measure of an antichain to its projections.

## Contribution

It introduces a new projection inequality for antichains, linking their Hausdorff measure to the measures of their projections, with implications for geometric measure theory.

## Key findings

- Hausdorff dimension of weak antichains is at most n-1
- Hausdorff measure of weak antichains is at most n
- Projection inequality bounds measure of antichains by projections

## Abstract

A set $A \subseteq {\mathbb{R}}^n$ is called an antichain (resp. antichain) if it does not contain two distinct elements ${\mathbf x}=(x_1,\ldots, x_n)$ and ${\mathbf y}=(y_1,\ldots, y_n)$ satisfying $x_i\le y_i$ (resp. $x_i < y_i$) for all $i\in \{1,\ldots,n\}$. 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.