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.

References in corpus (4)

• Polyhedra in loop quantum gravity
• Generalization of Sabitov's Theorem to Polyhedra of Arbitrary Dimensions
• Slices, slabs, and sections of the unit hypercube
• An Algebraic Identity Leading to Wilson Theorem