Deriving New Sinc Results from Old

From previously established results in [2] we develop a simple proof of Keith Ball’s expression in [1] for the volume of the intersection of an (n− 1)dimensional hyperplane with an n-dimensional cube, as well as a simple proof of the formula given by Frank and Riede in [5] for that volume. In our study in [2] of sinc integrals over a decade ago we established results—some of which were recapitulated in [4]—that lead by differentiation, as we now show, to a simple proof of K. Ball’s formula in [1] for the volume of the intersection of an (n− 1)-dimensional hyperplane with an n-dimensional cube (Ball’s original proof is itself quite direct), as well as a simple proof of the formula given very recently for that volume by Frank and Riede in [5]. We proved the following theorem in [2, Theorem 2, Remarks 1]. To assist in reading this note, we sketch that proof below. Theorem 1. Suppose that b > 0 and ak > 0 for k = 1, 2, . . . , n. For each of the 2 n ordered n-tuples γ := (γ1, γ2, . . . , γn) ∈ {−1, 1} define βγ = βγ(b) := b+ n ∑