Abstract. Box integrals—expectations 〈|⃗r | s 〉 or 〈|⃗r − ⃗q | s 〉 over the unit n-cube—have over three decades been occasionally given closed forms for isolated n, s. By employing experimental mathematics together with a new, global analytic strategy, we prove that for each of n = 1, 2, 3, 4dimensionsthebox integrals are for any integer s hypergeometrically closed (“hyperclosed”) in an explicit sense we clarify herein. For n = 5 dimensions, such a complete hyperclosure proof is blocked by a single, unresolved integral we call K5; although we do prove that all but a finite set of (n = 5) cases enjoy hyperclosure. We supply a compendium of exemplary closed forms that arise naturally from the theory. 1. Preliminaries We define box integrals 1 for positive-integer dimension n as expectations of |⃗r | s, |⃗r − ⃗q | s with the relevant vectors chosen randomly, independently, equidistributed over the unit n-cube2 [13]: Bn(s): = |⃗r | s D⃗r

Herein, with the aid of substantial symbolic computation, we solve previously open problems in the theory of n-dimensional box integrals Bn(s): = 〈|⃗r | s 〉; ⃗r ∈ [0, 1] n. In particular we resolve an elusive integral called K5 that previously acted as a blockade against closed-form evaluation in n = 5 dimensions. In consequence we now know that Bn (integer) can be given a closed form for n = 1, 2, 3, 4, 5. We also nd the general residue at the pole at s = −n, this leading to new relations and definite integrals for example, we are able to give the first nontrivial closed forms for 6-dimensional box integrals and to show hyperclosure of B6(even). The Clausen function and its generalizations play a central role in these higher-dimensional evaluations. Our results provide stringent test scenarios for symbolic-algebra simplification methods.

Using combinatorial methods, we derive several formulas for the volume of convex bodies obtained by intersecting a unit hypercube with a halfspace, or with a hyperplane of codimension 1, or with a flat defined by two parallel hyperplanes. We also describe some of the history of these problems, dating to Pólya’s Ph.D. thesis, and we discuss several applications of these formulas.

One of the most effective techniques of experimental mathematics is to compute mathematical entities such as integrals, series or limits to high precision, then attempt to recognize the resulting numerical values. Recently these techniques have been applied with great success to problems in mathematical physics. Notable among these applications are the identification of some key multi-dimensional integrals that arise in Ising theory, quantum field theory and in magnetic spin theory.