{
  "schema_version": 1,
  "problem_number": "AMR-022-4023",
  "title": "On a Question of Erdős on Lemniscates in R^m: Connected Lemniscates Lie in a Ball of Radius 2",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "For points x_1, …, x_n in R^m let E = {x ∈ R^m : ∏_{i=1}^n |x − x_i| ≤ 1}. A question of Erdős, posed in Hayman's collection of problems of 1974 and recorded as Problem 4.23 by Hayman and Lingham, asks how large the volume of E can be and whether a connected E can be covered by a ball of radius 2; for m = 2 the covering statement is a theorem of Pommerenke (1959). We answer the covering question: if E is connected, then |x − c| < 2 for every x ∈ E, where c is the centroid of x_1, …, x_n, for all m and n. Projecting E onto the line through c and a point of E reduces the question to Pommerenke's planar theorem for a polynomial with real zeros; we prove that theorem again, in the closed-set form needed. Collinear Chebyshev configurations give connected sets E with max_{x∈E} |x − c| = 2·2^(−1/n), so the constant 2 cannot be lowered uniformly in n, already for m = 1. The same projection shows that for each n the best radius about the centroid is the same in every dimension; it equals 2·2^(−1/n) for n ≤ 2, and whether it does for all n, as numerical experiments suggest, is open, as is the volume question. The question is Erdős's and the planar case is Pommerenke's; the note contributes the reduction, which gives the covering statement in every dimension and the dimension-independence of the best radius about the centroid. We found no proof of the covering statement for m ≥ 3 in the sources we searched. This is an unrefereed note.",
  "result_type": "COMPLETE_SCOPED_PROOF",
  "categories": [
    "math.CV",
    "math.MG"
  ],
  "keywords": [
    "lemniscates in Euclidean space",
    "products of distances",
    "connected sublevel sets",
    "covering by a ball",
    "Erdős problems",
    "Hayman–Lingham Problem 4.23",
    "Pommerenke's theorem",
    "polynomials with real zeros",
    "Chebyshev polynomials",
    "univalent functions",
    "partial answer",
    "UnsolvedMath",
    "AMR-022-4023",
    "math.CV",
    "math.MG",
    "open mathematics"
  ],
  "manuscript_version_date": "2026-10-16",
  "publication_date": "2026-10-16",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-10-16",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/amr-022-4023/",
  "pdf_url": "https://eulersolve.org/papers/amr-022-4023/paper.pdf?v=48d764cedfc2",
  "doi": "10.5281/zenodo.23422190",
  "zenodo_record_url": "https://zenodo.org/records/23422190",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "Answers the covering question of Problem 4.23 in every dimension (the planar case is Pommerenke's theorem); the volume question and the exact best radius for fixed n remain open.",
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    "source.zip": {
      "sha256": "d54f1832af18e162554120793adcb1d74d6637e9efdf7cc39aca698688f88a85"
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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."
}