Central diagonal sections of Gaussian $n$-cubes
Ferenc Fodor, Bernardo González Merino
TL;DR
The paper derives the exact asymptotic limit for the central diagonal section measure of the Gaussian-type distribution on the cube $C^n$, parameterized by $b>0$, extending the classical Lebesgue-case result. It expresses the central-section measure via an integral representation and analyzes it by transforming the inner integral into a form involving the error function, solving a related differential equation, and applying Taylor expansions together with a Catalan-number identity to sum the limit. The main finding is $\displaystyle \lim_{n\to\infty}A(a,\gamma_n[b])=2\sqrt{\frac{b}{\pi}}\left(1-4\frac{e^{-b}\sqrt{b}}{2\sqrt{\pi}\operatorname{erf}(\sqrt{b})}\right)^{-\frac{1}{2}}$, which recovers the Lebesgue case as $b\to0^+$. The work connects to prior results on maximal central sections and midpoints a rich lineage of convex-body geometry results, providing a concrete, closed-form asymptotic for Gaussian-weighted sections.
Abstract
The investigation of the volume, surface area, and other geometric properties of sections of convex bodies, and in particular cubes, has a long history and a rich literature. However, much less is known when the cube has a volume distribution that is different from the Lebesgue measure; for example, a Gaussian density. We study the probability densities in the unit cube $C^n=[-1,1]^n$ of $\mathbb R^n$ generated by $e^{-b\|x\|^2}$, $b> 0$. We prove that the limit of the induced Gaussian-type volume of sections of $C^n$ through the origin and orthogonal to a main diagonal is \[ \sqrt{\frac bπ}\left (1-4\frac{e^{-b}\sqrt{b}}{2\sqrtπ\mathrm{erf}(\sqrt{b})}\right)^{-\frac12}, \] as $n\to\infty$. This extends the well-known result of Hensley (1979) for the Lebesgue measure and continues the investigations initiated by Barthe, Guédon, Mendelson, Naor (2005), Zvavitch (2008), and König, Koldobski (2013). The proof uses a mixture of techniques from analysis and probability.