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, 1ᵈ is t. If t is equal to 0 and H is orthogonal to a diagonal of 0, 1ᵈ, it is known that the (d-1) -dimensional volume of H0, 1ᵈ 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 H0, 1ᵈ when H is orthogonal to a diagonal of either 0, 1ᵈ 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, 1ᵈ. A precise estimate for the convergence rate of the Eulerian numbers to their limit Gaussian behavior is provided along the way.