How to integrate a polynomial over a simplex

Baldoni, Velleda; Berline, Nicole; Loera, Jesús A. De; Köppe, Matthias; Vergne, Michèle

doi:10.1090/s0025-5718-2010-02378-6

Cited by 77 publications

(151 citation statements)

References 37 publications

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“…3.5] and in particular [5,Corollary 3.5.6]. A variation of (1.10) also appears in [6], in the context of designing an efficient procedure for integration of polynomials over simplices.…”

Section: Theorem 2 Impliesmentioning

confidence: 99%

On moments of a polytope

et al. 2018

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We show that the multivariate generating function of appropriately normalized moments of a measure with homogeneous polynomial density supported on a compact polytope P ⊂ R d is a rational function. Its denominator is the product of linear forms dual to the vertices of P raised to the power equal to the degree of the density function. Using this, we solve the inverse moment problem for the set of, not necessarily convex, polytopes having a given set S of vertices. Under a weak nondegeneracy assumption we also show that the uniform measure supported on any such polytope is a linear combination of uniform measures supported on simplices with vertices in S.

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“…3.5] and in particular [5,Corollary 3.5.6]. A variation of (1.10) also appears in [6], in the context of designing an efficient procedure for integration of polynomials over simplices.…”

Section: Theorem 2 Impliesmentioning

confidence: 99%

On moments of a polytope

et al. 2018

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“…In the remainder, however, we will study this domain as a subset of the generalized (Poincaré) upper half plane H rather than the unit hyperboloid. 2 This upper half plane is defined as…”

Section: Volume Formulamentioning

confidence: 99%

On Fundamental Domains and Volumes of Hyperbolic Coxeter–Weyl Groups

2011

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Abstract. We present a simple method for determining the shape of fundamental domains of generalized modular groups related to Weyl groups of hyperbolic Kac-Moody algebras. These domains are given as subsets of certain generalized upper half planes, on which the Weyl groups act via generalized modular transformations. Our construction only requires the Cartan matrix of the underlying finite-dimensional Lie algebra and the associated Coxeter labels as input information. We present a simple formula for determining the volume of these fundamental domains. This allows us to re-produce in a simple manner the known values for these volumes previously obtained by other methods.

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“…For the simplex one can find several contributions in the literature for integrating polynomials and defining cubatures formula; see for instance the recent work of Baldoni et al [2] and the many references therein. But concerning the slice of a simplex it turns out that a formula for the volume of the slice has already been derived ... as a by-product in the construction of univariate B-splines!…”

Section: Introductionmentioning

confidence: 99%

Volume of slices and sections of the simplex in closed form

Lasserre

2015

Optim Lett

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Abstract. Given a vector a ∈ R n , we provide an alternative and direct proof for the formula of the volume of sections ∆ ∩ {x : a T x <= t} and slices ∆ ∩ {x : a T x = t}, t ∈ R, of the simplex ∆. For slices the formula has already been derived but as a byproduct 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 .

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How to integrate a polynomial over a simplex

Cited by 77 publications

References 37 publications

On moments of a polytope

On moments of a polytope

On Fundamental Domains and Volumes of Hyperbolic Coxeter–Weyl Groups

Volume of slices and sections of the simplex in closed form

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