Volume of slices and sections of the simplex in closed form

TL;DR

This paper presents a new, straightforward proof for the volume formulas of slices and sections of a simplex, demonstrating the effectiveness of Laplace transform techniques for deriving closed-form multivariate integrals.

Contribution

It offers an alternative proof for simplex volume formulas and highlights the utility of Laplace transforms in multivariate integral calculations.

Findings

Provides a direct proof for volume of simplex slices and sections.

Shows Laplace transform as a powerful tool for multivariate integral solutions.

Complements previous results on hypercube volume calculations.

Abstract

Given a vector a $\in$ Rn, we provide an alternative and direct proof for the formula of the volume of sections delta $\cap$ {x : a T x \textless{}= t} and slices $\cap$ {x : a T x = t}, t $\in$ R, of the simplex delta. For slices the formula has already been derived but as a by-product of the construction of univariate B-Splines. One goal of the paper is to also show how simple and powerful can be the Laplace transform technique to derive closed form expression for some multivariate integrals. It also complements some previous results obtained for the hypercube [0, 1] n .