Central Diagonal Sections Of The N-Cube

We prove that the volume of central hyperplane sections of a unit cube in Rn orthogonal to a main diagonal of the cube is a strictly monotonically increasing function of the dimension for n >= 3. Our argument uses an integral formula that goes back to Polya [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.