Volume of Hypercubes Clipped by Hyperplanes and Combinatorial Identities

Abstract

There is an elegant expression for the volume of hypercube $[0,1]^n$ clipped by a single hyperplane. In the article, the formula is generalized to the case of more than one hyperplane. An important foundation for the result is Lawrence's formula and a way to weaken two restrictions of simplicity and non-parallelness in his formula is also considered. Several concrete volume formulas of clipped hypercubes are derived explicitly and the corresponding combinatorial identities are obtained as an application.