  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.
