Slices, Slabs, and Sections of the Unit Hypercube

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 P'{o}lya’s Ph.D. thesis, and we discuss several applications of these formulas.