Deep sections of the hypercube

Consider a non-negative number t and a hyperplane H of Rᵈ whose distance to the center of the hypercube [0,1]ᵈ is t. If t is equal to $0$ and H is orthogonal to a diagonal of [0,1]ᵈ, it is known that the $(d-1)$-dimensional volume of H∩[0,1]ᵈ is a strictly increasing function of d when d is at least $3$. The study of the monotonicity of this volume is extended for t up to above $1/2$ and, when d is large enough, for every non-negative t. In particular, a range for t is identified such that this volume is a strictly decreasing function of d over the positive integers. The local extremality of the $(d-1)$-dimensional volume of H∩[0,1]ᵈ when H is orthogonal to a diagonal of either [0,1]ᵈ or a lower dimensional face is also determined for the same values of t. It is shown for instance that when t is above an explicit constant and d is large enough, this volume is always strictly locally maximal when H is orthogonal to a diagonal of [0,1]ᵈ. A precise estimate for the convergence rate of the Eulerian numbers to their limit Gaussian behavior is provided along the way.