{
  "abstract": "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 $x\\prec_C y\\iff y-x\\in C$ 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) \\leq k\\frac{\\|v\\|}{α_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 $α_C(v)$ is the minimum of $\\langleξ,v\\rangle$ over unit vectors in the dual cone $C^*$. In particular, every weak antichain $A\\subseteq\\mathbb R^n$ satisfies $\\mathcal H^{n-1}(A)\\leq\\mathcal H^{n-1}(Δ_i(A))$ for each $i$, where $Δ_i(x)=(x_j-x_i)_{j\\ne i}$. This proves the skewed projection conjecture of Engel, Mitsis, Pelekis and Reiher. As an application, we recover the bound $\\mathcal H^{n-1}(A)\\leq kn$ for $k$-antichains in the unit cube under the coordinatewise order.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.14563",
  "author_total": 2,
  "authors": [
    {
      "name": "Yuxian Dong"
    },
    {
      "name": "Lili Mu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/3ppe4be7f4wb9.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.14563",
  "fulltext_url": null,
  "id": "3ppe4be7f4wb9",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.14563",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ],
    [
      "openalex",
      "2607|Advanced Combinatorial Mathematics"
    ]
  ],
  "openalex": "W7221807963",
  "pdf_url": "https://arxiv.org/pdf/2610.14563",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-10",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/3ppe4be7f4wb9.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:15Z",
      "native_id": "2610.14563",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.14563"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Sharp projection inequalities for antichains in cone orders",
  "title_text": "Sharp projection inequalities for antichains in cone orders",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:15Z",
  "url": "https://research.draconitic.com/paper/3ppe4be7f4wb9",
  "venue": "arXiv",
  "versions": []
}