Central Diagonal Sections of the n-Cube

We prove that the volume of central hyperplane sections of a unit cube in $\mathbb{R}^n$ orthogonal to a main diagonal of the cube is a strictly monotonically increasing function of the dimension for $n\geq 3$. Our argument uses an integral formula that goes back to Pólya [ 20] (see also [ 14] and [ 3]) for the volume of central sections of the cube and Laplace’s method to estimate the asymptotic behavior of the integral. First, we show that monotonicity holds starting from some specific $n_0$. Then, using interval arithmetic and automatic differentiation, we compute an explicit bound for $n_0$ and check the remaining cases between $3$ and $n_0$ by direct computation.