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 $n\geq2$ and $k\ge 1$, let $C\subsetneq\mathbb R^n$ 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\iff y-x\in