Projections of Antichains

A subset $A$ of $\mathbb{Z}^n$ is called a weak antichain if it does not contain two elements $x$ and $y$ satisfying $x_i<y_i$ for all $i$. Engel, Mitsis, Pelekis and Reiher showed that for any weak antichain $A$, the sum of the sizes of its $(n-1)$-dimensional projections must be at least as large as its size $|A|$. They asked what the smallest possible value of the gap between these two quantities is in terms of $|A|$. We answer this question by giving an explicit weak antichain attaining this minimum for each possible value of $|A|$. In particular, we show that sets of the form $$A_N=\{x\in\mathbb{Z}^n: 0\leq x_j\leq N-1 \textrm{ for all $j$ and } x_i=0\textrm{ for some $i$}\}$$ minimise the gap among weak antichains of size $|A_N|$.