Central diagonal sections of Gaussian cubes

Ferenc Fodor; Bernardo González Merino

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arXiv · 2511.01504

Central diagonal sections of Gaussian cubes

Ferenc Fodor·Bernardo González Merino

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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 standard cube $B^n_\infty=[-1,1]^n$ of $\mathbb R^n$ generated by $e^{-b\|x\|^2}$, $b> 0$. We prove that the limit of the induced Gaussian-type volume of hyperplane sections of $B^n_\infty$ 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).

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Ferenc Fodor, Bernardo González Merino. 2025-11-03. Central diagonal sections of Gaussian cubes. https://arxiv.org/abs/2511.01504

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