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 \(K\subset\mathbb{R}^d\). Meckes asked, as recorded by Reitzner in Oberwolfach Report 10/2017, whether there is a constant \(c(d)\) with \(\lim_dc(d)^{1/d}<\infty\) such that \(\mathbb{E} V_K\le c(d)\,\mathbb{E} V_L\) whenever \(K\subseteq L\), 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 \(c^*(d)=\sup_{K\subseteq L}\mathbb{E} V_K/\mathbb{E} V_L\). 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 \(c^*(d)\le\theta_d(L_d/L_{B^d})^d\) with \(\theta_d\le((d+1)\binom{d+8}{8})^{1/2}\), where \(L_d\) is the largest isotropic constant in \(\mathbb{R}^d\) and \(B^d\) is the unit ball; hence \(\limsup c^*(d)^{1/d}\le\sqrt{2\pi e}\,\sup_nL_n\). For the second question we prove that \(c^*(d)\) grows exponentially: \(\liminf_dc^*(d)^{1/d}\ge(300657950289351/292457950289351)^{1/2}=1.01392\ldots\), so that \(c^*(d)=e^{\Theta(d)}\). Explicitly, \(c^*(d)\ge((d+1)\binom{d+27}{27})^{-1/2}\,1.28257669^{\lfloor d/9\rfloor/2}\) for every \(d\); this exceeds \(1\) only for \(d\ge6570\) (a sharper product form only for \(d\ge6525\)), so for smaller \(d\) it is weaker than the trivial bound \(c^*(d)\ge1\), and the asymptotic statement is the main result. The proof combines Rademacher's identity \(\mathbb{E} V_K^2=\frac{d+1}{d!}\det\operatorname{Cov}(K)\), a Gram–Schmidt and Hölder argument showing that for products of copies of a fixed body \(\mathbb{E} V_K\) is within a polynomial factor of \((\mathbb{E} V_K^2)^{1/2}\), and an explicit pair \(K_0\subseteq L_0\) in \(\mathbb{R}^9\) with \(\det\operatorname{Cov}(K_0)>\det\operatorname{Cov}(L_0)\), 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.