{
  "schema_version": 1,
  "problem_number": "OWR-15215-006",
  "title": "On Meckes's Monotonicity Question for the Expected Volume of a Random Simplex: The Optimal Constant Grows Exponentially",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "Let V_K be the volume of the simplex spanned by d+1 independent uniform random points in a convex body K ⊂ ℝ^d. Meckes asked, as recorded by Reitzner in Oberwolfach Report 10/2017, whether there is a constant c(d) with lim_d c(d)^(1/d) < ∞ such that E V_K ≤ c(d) E V_L whenever K ⊆ L, and the report asks for a lower bound for c(d). The displayed definition of c(d) as a minimum is degenerate if read literally; we read the second question as asking for lower bounds for the optimal constant c*(d) = sup_{K⊆L} E V_K / E V_L. The answer to the first question is yes, and it is in print: Rademacher proved that it is equivalent to the slicing problem (an equivalence also stated by Reichenwallner and Reitzner), which Klartag and Lehec solved. We add a short self-contained proof, based on Groemer's theorem, of the explicit bound c*(d) ≤ θ_d (L_d / L_{B^d})^d with θ_d ≤ ((d+1) binom(d+8, 8))^(1/2), where L_d is the largest isotropic constant in ℝ^d and B^d is the unit ball; hence limsup c*(d)^(1/d) ≤ √(2πe) sup_n L_n. For the second question we prove that c*(d) grows exponentially: liminf_d c*(d)^(1/d) ≥ (300657950289351/292457950289351)^(1/2) = 1.01392…, so that c*(d) = e^{Θ(d)}. Explicitly, c*(d) ≥ ((d+1) binom(d+27, 27))^(−1/2) · 1.28257669^(⌊d/9⌋/2) for every d; this exceeds 1 only for d ≥ 6570 (a sharper product form only for d ≥ 6525), so for smaller d it is weaker than the trivial bound c*(d) ≥ 1, and the asymptotic statement is the main result. The proof combines Rademacher's identity E V_K² = ((d+1)/d!) det Cov(K), a Gram–Schmidt and Hölder argument showing that for products of copies of a fixed body E V_K is within a polynomial factor of (E V_K²)^(1/2), and an explicit pair K_0 ⊆ L_0 in ℝ^9 with det Cov(K_0) > det Cov(L_0), certified in exact rational arithmetic: L_0 is the convex hull of a regular simplex and the apexes of pyramids of height 5/4 (in units of the inradius) on its facets, and K_0 keeps of each pyramid only the bottom 1/11 of its height. No earlier quantitative lower bound for c*(d) was found in the sources we searched, and the product idea may be folklore. The exact value of c*(d) and the limit of c*(d)^(1/d) remain open. This is an unrefereed note.",
  "result_type": "COMPLETE_SCOPED_PROOF",
  "categories": [
    "math.MG",
    "math.PR",
    "math.FA"
  ],
  "keywords": [
    "random simplex",
    "expected volume",
    "monotonicity under inclusion",
    "Meckes's weak conjecture",
    "slicing problem",
    "isotropic constant",
    "covariance matrix",
    "Groemer's theorem",
    "product bodies",
    "exponential lower bound",
    "Oberwolfach Reports open problems",
    "UnsolvedMath",
    "OWR-15215-006",
    "math.MG",
    "math.PR",
    "math.FA",
    "open mathematics"
  ],
  "manuscript_version_date": "2026-10-15",
  "publication_date": "2026-10-15",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-10-15",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/owr-15215-006/",
  "pdf_url": "https://eulersolve.org/papers/owr-15215-006/paper.pdf?v=a021c9dbef4f",
  "doi": "10.5281/zenodo.23392932",
  "zenodo_record_url": "https://zenodo.org/records/23392932",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "On Meckes's question recorded by Reitzner (Oberwolfach Report 10/2017): the existence of a constant with at most exponential growth is in print (Rademacher together with Klartag and Lehec), and the optimal constant c*(d) itself grows exponentially, c*(d) = exp(Theta(d)); the explicit lower bound exceeds 1 only for d at least 6525, and the exact constant remains open.",
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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."
}