A generalization of Filliman duality

Author:
Greg Kuperberg

Journal:
Proc. Amer. Math. Soc. **131** (2003), 3893-3899

MSC (2000):
Primary 52B45

DOI:
https://doi.org/10.1090/S0002-9939-03-06957-0

Published electronically:
February 28, 2003

MathSciNet review:
1999938

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Abstract | References | Similar Articles | Additional Information

Abstract: Filliman duality expresses (the characteristic measure of) a convex polytope containing the origin as an alternating sum of simplices that share supporting hyperplanes with . The terms in the alternating sum are given by a triangulation of the polar body . The duality can lead to useful formulas for the volume of . A limiting case called Lawrence's algorithm can be used to compute the Fourier transform of .

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 is a convex polytope containing the origin, any realization of as a linear combination of simplices leads to a dual realization of .

**1.**Alexander Barvinok,*Convexity*, to appear in Graduate Studies in Mathematics, 2002.**2.**Paul Filliman,*The volume of duals and sections of polytopes*, Mathematika**39**(1992), no. 1, 67-80. MR**93g:52005****3.**Jim Lawrence,*Valuations and polarity*, Discrete Comput. Geom.**3**(1988), no. 4, 307-324. MR**90b:52001****4.**-,*Polytope volume computation*, Math.Comp.**57**(1991), no. 195, 259-271. MR**91j:52019****5.**W. B. R. Lickorish,*Simplicial moves on complexes and manifolds*, Geom. Topol. Monogr.**2**(1999), 299-320, arXiv:math.GT/9911256. MR**2000j:57057****6.**Peter McMullen,*The polytope algebra*, Adv. Math.**78**(1989), no. 1, 76-130. MR**91a:52017****7.**M. H. A. Newman,*A theorem in combinatorial topology*, J. London Math. Soc.**6**(1931), 186-192.

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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:
https://doi.org/10.1090/S0002-9939-03-06957-0

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