Sharp projection inequalities for antichains in cone orders
Abstract
We establish sharp projection inequalities for k-antichains under orders defined by open convex cones. For integers n≥2 and k≥ 1, let Csubsetneqℝn be a nonempty open convex cone. A set A⊆ℝn is a k-antichain with respect to the order x≺C y⟺ y−x∈ C if it contains no chain x1≺C⋯≺C xk+1. For every such set A and every v∈ C, we prove ℋn−1(A) ≤ k|v|/αC(v) ℋn−1(Pv(A)), where Pv is the orthogonal projection onto v⊥, ℋn−1 denotes normalized Hausdorff outer measure, and αC(v) is the minimum of ⟨ξ,v⟩ over unit vectors in the dual cone C*. In particular, every weak antichain A⊆ℝn satisfies ℋn−1(A)≤ℋn−1(Δi(A)) for each i, where Δi(x)=(xj−xi)j≠ i. This proves the skewed projection conjecture of Engel, Mitsis, Pelekis and Reiher. As an application, we recover the bound ℋn−1(A)≤ kn for k-antichains in the unit cube under the coordinatewise order.
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BibTeX
@misc{dong2026sharp,
title = {{Sharp projection inequalities for antichains in cone orders}},
author = {Yuxian Dong and Lili Mu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.14563},
eprint = {2610.14563},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.14563},
}
RIS
TY - UNPB
TI - Sharp projection inequalities for antichains in cone orders
AU - Yuxian Dong
AU - Lili Mu
PY - 2026
DA - 2026/10/10
PB - arXiv
DO - 10.48550/arxiv.2610.14563
UR - https://arxiv.org/abs/2610.14563
AB - We establish sharp projection inequalities for k-antichains under orders defined by open convex cones. For integers n≥2 and k≥ 1, let Csubsetneqℝ^n be a nonempty open convex cone. A set A⊆ℝ^n is a k-antichain with respect to the order x≺_C y⟺ y−x∈ C if it contains no chain x^1≺_C⋯≺_C x^k+1. For every such set A and every v∈ C, we prove ℋ^n−1(A) ≤ k|v|/α_C(v) ℋ^n−1(P_v(A)), where P_v is the orthogonal projection onto v^⊥, ℋ^n−1 denotes normalized Hausdorff outer measure, and α_C(v) is the minimum of ⟨ξ,v⟩ over unit vectors in the dual cone C^*. In particular, every weak antichain A⊆ℝ^n satisfies ℋ^n−1(A)≤ℋ^n−1(Δ_i(A)) for each i, where Δ_i(x)=(x_j−x_i)_j≠ i. This proves the skewed projection conjecture of Engel, Mitsis, Pelekis and Reiher. As an application, we recover the bound ℋ^n−1(A)≤ kn for k-antichains in the unit cube under the coordinatewise order.
ER -