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2020•Electronic Journal of Linear AlgebraOpen access

Volume of Hypercubes Clipped by Hyperplanes and Combinatorial Identities

Yunhi Cho, Seonhwa Kim

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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.

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What this paper is about

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.

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Available 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.

Key concepts: Hyperplane, Hypercube, Mathematics, Volume (thermodynamics), Combinatorics, Simplicity, Discrete mathematics, Physics

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