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NEIGHBORLINESS OF THE SYMMETRIC MOMENT CURVE

Part of: Polynomials, rational functions Polytopes and polyhedra General convexity

Published online by Cambridge University Press:  12 April 2012

Alexander Barvinok ,
Seung Jin Lee  and
Isabella Novik
Alexander Barvinok
Affiliation:
Department of Mathematics, University of Michigan, Ann Arbor, MI 48109-1043, U.S.A. (email: barvinok@umich.edu)
Seung Jin Lee
Affiliation:
Department of Mathematics, University of Michigan, Ann Arbor, MI 48109-1043, U.S.A. (email: lsjin@umich.edu)
Isabella Novik
Affiliation:
Department of Mathematics, University of Washington, Seattle, WA 98195-4350, U.S.A. (email: novik@math.washington.edu)
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Abstract

We consider the convex hull ℬk of the symmetric moment curve Uk(t)=(cos t,sin t,cos 3t,sin 3t,…,cos (2k−1)t,sin (2k−1)t) in ℝ2k, where t ranges over the unit circle 𝕊=ℝ/2πℤ. The curve Uk(t) is locally neighborly: as long as t1,…,tk lie in an open arc of 𝕊 of a certain length ϕk>0 , the convex hull of the points Uk (t1),…,Uk (tk) is a face of ℬk. We characterize the maximum possible length ϕk, proving, in particular, that ϕk >π/2 for all k and that the limit of ϕk is π/2 as k grows. This allows us to construct centrally symmetric polytopes with a record number of faces.

MSC classification

Secondary: 52A20: Convex sets in $n$ dimensions (including convex hypersurfaces) 52B12: Special polytopes (linear programming, centrally symmetric, etc.) 26C10: Polynomials
Type
Research Article
Information
Mathematika , Volume 59 , Issue 1 , January 2013 , pp. 223 - 249
Copyright
Copyright © University College London 2012

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