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Integration on a convex polytope


Author: Jean B. Lasserre
Journal: Proc. Amer. Math. Soc. 126 (1998), 2433-2441
MSC (1991): Primary 65D30
DOI: https://doi.org/10.1090/S0002-9939-98-04454-2
MathSciNet review: 1459132
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Abstract: We present an exact formula for integrating a (positively) homogeneous function $f$ on a convex polytope $\Omega\subset R^n$. We show that it suffices to integrate the function on the $(n-1)$-dimensional faces of $\Omega$, thus reducing the computational burden. Further properties are derived when $f$ has continuous higher order derivatives. This result can be used to integrate a continuous function after approximation via a polynomial.


References [Enhancements On Off] (What's this?)

  • 1. Alexander I. Barvinok, Computing the volume, counting integral points, and exponential sums, Discrete Comput. Geom. 10 (1993), no. 2, 123–141. MR 1220543, https://doi.org/10.1007/BF02573970
  • 2. Michel Brion, Points entiers dans les polyèdres convexes, Ann. Sci. École Norm. Sup. (4) 21 (1988), no. 4, 653–663 (French). MR 982338
  • 3. B. Bueeler, A. Enge, K. Fukuda, H-J. Lthi, Exact volume computation for polytopes: a practical study, 12th European Workshop on Computational Geometry, Muenster, Germany, March 1996.
  • 4. J. B. Lasserre, An analytical expression and an algorithm for the volume of a convex polyhedron in 𝑅ⁿ, J. Optim. Theory Appl. 39 (1983), no. 3, 363–377. MR 703477, https://doi.org/10.1007/BF00934543
  • 5. G. Cagnac, E.Ramis, J. Commeau, Analyse, Masson, Paris, 1970.
  • 6. M.E. Taylor, Partial Differential Equations: Basic Theory, Springer-Verlag, New York, 1996. CMP 96:14

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Additional Information

Jean B. Lasserre
Affiliation: LAAS-CNRS, 7 Avenue du Colonel Roche, 31077 Toulouse Cédex 4, France
Email: lasserre@laas.fr

DOI: https://doi.org/10.1090/S0002-9939-98-04454-2
Keywords: Numerical integration in $R^n$, homogeneous functions, convex polytopes
Received by editor(s): August 5, 1996
Received by editor(s) in revised form: January 6, 1997
Communicated by: David Sharp
Article copyright: © Copyright 1998 American Mathematical Society