On Meckes's Monotonicity Question for the Expected Volume of a Random Simplex: The Optimal Constant Grows Exponentially

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) = supK⊆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 / LB^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) = eΘ(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. Unrefereed preprint released for independent mathematical scrutiny. Publication on Zenodo does not constitute peer review. AI-assisted tools supported research, computation, proof development, and manuscript preparation. The author remains responsible for all claims and the final text. Corpus identifier: OWR-15215-006 (Meckes's question as recorded by M. Reitzner, High-dimensional random polytopes, Oberwolfach Report 10/2017, p. 519).