{
  "abstract": "Call a finite nonempty point set $S\\subset\\mathbb{R}^d$ all-even if for every direction $\\ell \\in \\mathbb{P}(\\mathbb{R}^d)$, the number of hyperplanes perpendicular to $\\ell$ that intersect $S$ is even. In 2026, Francisco Santos presented a question of Juan Benigno Seoane-Sepúlveda on the existence of all-even sets in dimensions $3$ and higher, together with examples of such sets in dimension $2$. Call a set $S\\in\\mathbb{R}^d$ line-all-even if for every direction $\\ell \\in \\mathbb{P}(\\mathbb{R}^d)$ the number of lines parallel to $\\ell$ that intersect $S$ is even. As Santos pointed out, every all-even set in $\\mathbb{R}^3$ is line-all-even. We conjecture that there exist no line-all-even sets in $\\mathbb{R}^3$ that affinely span $\\mathbb{R}^3$ We show that our conjecture is equivalent to the statement that every planar all-even set satisfies a certain property, which we call height-rigidity. This property has a natural interpretation in linear algebra terms. Furthermore, we present a hierarchy of properties of planar point sets that imply height-rigidity. We construct several all-even sets in the plane, including some infinite families, and show that all the sets we constructed are height-rigid. In addition, we prove that every all-even set in the plane that can be covered by six or fewer parallel lines is height-rigid.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.38864",
  "author_total": 2,
  "authors": [
    {
      "name": "Daniel Kalmanovich"
    },
    {
      "name": "Gabriel Nivasch"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/dpy6wyacz8amt.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.38864",
  "fulltext_url": null,
  "id": "dpy6wyacz8amt",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.38864",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.CG"
    ],
    [
      "arxiv",
      "math.CO"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2610.38864",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-28",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/dpy6wyacz8amt.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:36:32Z",
      "native_id": "2610.38864",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.38864"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "All-even sets in $\\mathbb{R}^2$ and $\\mathbb{R}^3$",
  "title_text": "All-even sets in ℝ^2 and ℝ^3",
  "type": "preprint",
  "updated_at": "2026-11-01T12:20:50Z",
  "url": "https://research.draconitic.com/paper/dpy6wyacz8amt",
  "venue": "arXiv",
  "versions": []
}