Abstract
Given a vector \(\mathbf {a}\in \mathbb {R}^n\), we provide an alternative and direct proof for the formula of the volume of sections \(\Delta \cap \{\mathbf {x}: \mathbf {a}^T\mathbf {x}<= t\}\) and slices \(\Delta \cap \{\mathbf {x}:\mathbf {a}^T\mathbf {x}= t\}\), \(t\in \mathbb {R}\), 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 the Laplace transform technique can be to derive closed form expressions for some multivariate integrals. It also complements some previous results obtained for the hypercube \([0,1]^n\).
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References
Barvinok, A.: Computing the volume, counting integral points, and exponential sums. Discrete. Comput. Geom. 10, 123–141 (1993)
Baldoni, V., Berline, N., De Loera, J., Köppe, M., Vergne, M.: How to integrate a polynomial over a simplex. Math. Comp. 80, 297–325 (2011)
Borwein, D., Borwein, J.M., Mares Jr, B.A.: Multi-variable sinc integrals and volumes of polyhedra. Ramanujan. J. 6, 189–208 (2002)
Curry, H.B., Schoenberg, I.J.: On Pólya frequency functions IV: the fundamental spline functions and their limits. J. Anal. Math. 17, 71–107 (1966)
Freitag, E., Busam, R.: Complex analysis, 2nd edn. Springer, Berlin (2009)
Lasserre, J.B.: An analytical expression and an algorithm for the volume of a convex polytope in \(\mathbb{R}^n\). J. Optim. Theory. Appl. 39, 363–377 (1983)
Lasserre, J.B., Zeron, E.S.: Solving a class of multivariate integration problems via Laplace techniques. Appl. Math. (Warsaw) 28, 391–405 (2001)
Marichal, J.L., Mossinghoff, M.J.: Slices, slabs, and sections of the unit hypercube, online. J. Anal. Comb. 3, 1 (2008)
Micchelli, C.A.: Mathematical aspects of geometric modeling. In: Proceedings of CBMS-NSF regional conference series, SIAM, Philadelphia (1995)
Pólya, G.: On a few questions in probability theory and some definite integrals related to them, PhD Thesis, Eötvös Loránd University (1912)
Widder, D.V.: The Laplace Transform. Princeton University Press, Princeton (1946)
Zhiqiang, Xu: Multivariate splines and polytopes. J. Approx. Theory. 163, 377–387 (2011)
Acknowledgments
The author declares that there is no conflict of interest. The work of the author has been partially supported by a PGMO grant from the Fondation Mathématique Jacques Hadamard (Paris).
