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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

A generalization of Filliman duality


Author: Greg Kuperberg
Journal: Proc. Amer. Math. Soc. 131 (2003), 3893-3899
MSC (2000): Primary 52B45
Published electronically: February 28, 2003
MathSciNet review: 1999938
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Abstract: Filliman duality expresses (the characteristic measure of) a convex polytope $P$ containing the origin as an alternating sum of simplices that share supporting hyperplanes with $P$. The terms in the alternating sum are given by a triangulation of the polar body $P^{\circ}$. The duality can lead to useful formulas for the volume of $P$. A limiting case called Lawrence's algorithm can be used to compute the Fourier transform of $P$.

In this note we extend Filliman duality to an involution on the space of polytopal measures on a finite-dimensional vector space, excluding polytopes that have a supporting hyperplane coplanar with the origin. As a special case, if $P$ is a convex polytope containing the origin, any realization of $P^{\circ}$ as a linear combination of simplices leads to a dual realization of $P$.


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Additional Information

Greg Kuperberg
Affiliation: Department of Mathematics, University of California-Davis, Davis, California 95616
Email: greg@math.ucdavis.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-03-06957-0
PII: S 0002-9939(03)06957-0
Received by editor(s): December 23, 2001
Received by editor(s) in revised form: June 24, 2002, and July 10, 2002
Published electronically: February 28, 2003
Additional Notes: The author was supported by NSF grant DMS #0072342
Communicated by: John R. Stembridge
Article copyright: © Copyright 2003 American Mathematical Society