Remote Access Proceedings of the American Mathematical Society
Green Open Access

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
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 $P$.

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

  • 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.

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 52B45

Retrieve articles in all journals with MSC (2000): 52B45

Additional Information

Greg Kuperberg
Affiliation: Department of Mathematics, University of California-Davis, Davis, California 95616

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

American Mathematical Society