Central diagonal sections of Gaussian $n$-cubes
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866909907672891392 |
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| author | Fodor, Ferenc Merino, Bernardo González |
| author_facet | Fodor, Ferenc Merino, Bernardo González |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2511_01504 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Central diagonal sections of Gaussian $n$-cubes Fodor, Ferenc Merino, Bernardo González Metric Geometry Functional Analysis Probability 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. |
| title | Central diagonal sections of Gaussian $n$-cubes |
| topic | Metric Geometry Functional Analysis Probability |
| url | https://arxiv.org/abs/2511.01504 |