{
  "schema_version": 1,
  "problem_number": "OWR-12723-008",
  "title": "The Sharp Hexagonal Constant in Steinerberger's Geometric Uncertainty Principle on the Square",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "For a partition of [0,1]² into N measurable pieces Ω_i, let F = Σ_i |Ω_i| A(Ω_i) + Σ_i (|Ω_i| − min_j |Ω_j|), where A is the Fraenkel asymmetry with respect to discs. Steinerberger proved F ≥ 1/60000 for large N and asked, in Oberwolfach Report 49/2013, whether the extremal configuration is a hexagonal tiling, which would give the optimal constant c ≈ 0.07. We show that this is so on the square, in an asymptotic sense, with the constant c_H = 0.07446575455…, the asymmetry of the regular hexagon. Every partition of [0,1]², into any number of pieces, satisfies F > c_H + 0.06 m, where m is the smallest measure of a piece, while for every N a honeycomb partition with a boundary correction has F ≤ c_H + O(N^(−1/2)). Hence the infimum I_N over partitions into N pieces lies between c_H + 0.03/N and c_H + O(N^(−1/2)), I_N tends to c_H as N → ∞, and the value c_H is attained by no partition. We do not show that near-minimisers are geometrically close to a honeycomb, and we do not treat other domains. The proof bounds F below by a covering functional of discs of unequal sizes and uses Laguerre cells, a moment lemma of Fejes Tóth for the indicator of the complement of a disc (proved in an appendix), Euler's formula, and a tangent inequality in the number of sides with explicit constants. The same proof answers, in the same asymptotic sense, a weighted version asked by Burchard: the limit of the infimum is min{α c_H, 1}, so honeycomb partitions stop being asymptotically extremal exactly at the weight α = 1/c_H = 13.4289…. This is an unrefereed note.",
  "result_type": "COMPLETE_PROOF",
  "categories": [
    "math.MG",
    "math.CA"
  ],
  "keywords": [
    "geometric uncertainty principle",
    "Fraenkel asymmetry",
    "hexagonal tiling",
    "honeycomb",
    "Fejes Tóth moment lemma",
    "Laguerre cells",
    "power diagram",
    "Oberwolfach Reports",
    "OWR-12723-008",
    "math.MG",
    "math.CA",
    "open mathematics"
  ],
  "manuscript_version_date": "2026-10-02",
  "publication_date": "2026-10-02",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-10-02",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/owr-12723-008/",
  "pdf_url": "https://eulersolve.org/papers/owr-12723-008/paper.pdf?v=e334ed209182",
  "doi": "10.5281/zenodo.23107228",
  "zenodo_record_url": "https://zenodo.org/records/23107228",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "Answers Steinerberger's question (OWR 49/2013, pp. 2891-2892) in the affirmative on the unit square, in an asymptotic sense: every partition of [0,1]^2 has F > c_H + 0.06 m, with c_H the Fraenkel asymmetry of the regular hexagon (0.0744657545...), and honeycomb partitions with a boundary correction reach c_H + O(N^(-1/2)), so the sharp constant is c_H and no partition attains it. The method is classical (Fejes Toth's moment lemma, Laguerre cells, Euler's formula; precedents Bourne-Peletier-Theil and Bourne-Cristoferi). Steinerberger's universal constant for arbitrary domains, the minimisers for a given N and the geometric closeness of near-minimisers to a honeycomb are not treated; a weighted version asked by Burchard is answered in the same asymptotic sense (threshold 1/c_H).",
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    "source.zip": {
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    "verification_report.md": {
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  "ai_use_disclosure": "AI-assisted tools supported research, computation, proof development, and manuscript preparation. The author remains responsible for all claims and the final text."
}