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
Polya's Ph.D. thesis, and we discuss several applications of these formulas.Comment: 11 pages; minor corrections to reference
