{
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  "slug": "almost-equilateral-banach-logarithmic-dimension-status-20260801",
  "type": "claim",
  "title": "Current checked status and unresolved remainder",
  "summary": "UNKNOWN as of 2026-07-27. The MathOverflow page has zero answers. Checked literature gives the logarithmic bound for classical l_p spaces and a general log-squared-type bound, while no source located in the dated search establishes the stated uniform logarithmic bound for all norms.",
  "relevance": "Records the strongest checked neighboring results and the exact remainder future work must settle.",
  "relevance_source": "recorded",
  "body": "A dated independent review on 2026-08-01 checked the structured sources below, the complete visible source discussion, exact-title and equivalent-formulation searches, and the current TheoremDB corpus. On 2026-07-27 the Stack Exchange API reported zero answers, no accepted answer, and no closure for question 188161; comments identify the general log-squared bound as the strongest known there. Bartal, Linial, Mendel, and Naor, European Journal of Combinatorics 25 (2004), prove exponential-size almost-equilateral sets for the classical l_p families, uniformly over p, rather than arbitrary Banach spaces. Arias-de-Reyna, Ball, and Villa, Mathematika 45 (1998), give related large almost-equilateral configurations but leave a quantitative gap from the target. A TheoremDB search for almost-equilateral Banach sets, logarithmic dimension, and universal normed-space embeddings found no duplicate.\n\nA complete resolution must satisfy: Prove the existence of C(epsilon) with the displayed universal quantifiers, or construct a fixed epsilon and a sequence of d-dimensional Banach spaces whose largest epsilon-almost-equilateral sets have subexponential size in d. A quantitative result must state all dependencies on epsilon and may rescale a constructed set only by a single common factor.",
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    "citation": {
      "url": "https://mathoverflow.net/questions/188161/large-almost-equilateral-sets-in-finite-dimensional-banach-spaces",
      "locator": "Dataset references and independent 2026-08-01 status search."
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  "source": {
    "url": "https://mathoverflow.net/questions/188161/large-almost-equilateral-sets-in-finite-dimensional-banach-spaces",
    "locator": "Dataset references and independent 2026-08-01 status search."
  },
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      "slug": "almost-equilateral-banach-logarithmic-dimension",
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