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 on a convex polytope . We show that it suffices to integrate the function on the -dimensional faces of , thus reducing the computational burden. Further properties are derived when has continuous higher order derivatives. This result can be used to integrate a continuous function after approximation via a polynomial.

**1.**A. Barvinok,*Computing the volume, counting integral points, and exponential sums*, Discrete & Computational Geometry 10 (1993), pp. 123-141. MR**94d:52005****2.**M. Brion,*Points entiers dans les polydres convexes*, Ann. Sci. Ec. Norm. Sup., Série IV, 21 (1988), pp. 653-663. MR**90d:52020****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*, J. Optim. Theor. Appl. 39 (1983), pp. 363-377. MR**84m:52018****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