Central diagonal sections of Gaussian cubes

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 $B_{\mathrm{\infty }}^n=[-1,1{]}^n$ of ${\mathbb{R}}^n$ generated by $e^{-b\mathrm{\parallel }x{\mathrm{\parallel }}^2}$, $b>0$. We prove that the limit of the induced Gaussian-type volume of hyperplane sections of $B_{\mathrm{\infty }}^n$ through the origin and orthogonal to a main diagonal is $$\sqrt{\frac{b}{\pi }}{(1-4\frac{e^{-b}\sqrt{b}}{2\sqrt{\pi }\mathrm{e}\mathrm{r}\mathrm{f}(\sqrt{b})})}^{-\frac{1}{2}},$$ as $n\to \mathrm{\infty }$. 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).