Central diagonal sections of Gaussian $n$-cubes

TL;DR

This paper examines the asymptotic volume of Gaussian-weighted sections of high-dimensional cubes, extending classical results for Lebesgue measure to Gaussian measures and employing analysis and probability techniques.

Contribution

It derives the limit of Gaussian-type volume of diagonal sections of high-dimensional cubes, generalizing Hensley's classical Lebesgue measure result.

Findings

Limit of Gaussian volume of sections as dimension tends to infinity

Extension of classical Lebesgue measure results to Gaussian measures

Uses analysis and probability techniques in proof

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 \[ \sqrt{\frac b\pi}\left (1-4\frac{e^{-b}\sqrt{b}}{2\sqrt{\pi}\mathrm{erf}(\sqrt{b})}\right)^{-\frac12}, \] as $n\to\infty$. This extends the well-known result of Hensley (1979) for the Lebesgue measure and continues the investigations initiated by Barthe, Gu\'edon, Mendelson, Naor (2005), Zvavitch (2008), and K\"onig, Koldobski (2013). The proof uses a mixture of techniques from analysis and probability.