Sharp projection inequalities for antichains in cone orders

We establish sharp projection inequalities for $k$-antichains under orders defined by open convex cones. For integers $n\ge 2$ and $k\ge 1$, let $C⊊{\mathbb{R}}^n$ be a nonempty open convex cone. A set $A\subseteq {\mathbb{R}}^n$ is a $k$-antichain with respect to the order $x{\prec }_{C}y⟺y-x\in C$ if it contains no chain $x^1{\prec }_C\cdots {\prec }_C{x}^{k+1}$. For every such set $A$ and every $v\in C$, we prove $${\mathcal{H}}^{n-1}(A)\le k\frac{\mathrm{\parallel }v\mathrm{\parallel }}{{\alpha }_C(v)}{\mathcal{H}}^{n-1}(P_v(A)),$$ where $P_v$ is the orthogonal projection onto $v^{\perp }$, ${\mathcal{H}}^{n-1}$ denotes normalized Hausdorff outer measure, and ${\alpha }_C(v)$ is the minimum of $⟨\xi ,v⟩$ over unit vectors in the dual cone $C^{\ast }$. In particular, every weak antichain $A\subseteq {\mathbb{R}}^n$ satisfies ${\mathcal{H}}^{n-1}(A)\le {\mathcal{H}}^{n-1}({\mathrm{\Delta }}_i(A))$ for each $i$, where ${\mathrm{\Delta }}_i(x)=(x_j-x_i{)}_{j\mathrm{\ne }i}$. This proves the skewed projection conjecture of Engel, Mitsis, Pelekis and Reiher. As an application, we recover the bound ${\mathcal{H}}^{n-1}(A)\le kn$ for $k$-antichains in the unit cube under the coordinatewise order.