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NEIGHBORLINESS OF THE SYMMETRIC MOMENT CURVE

  • Alexander Barvinok (a1), Seung Jin Lee (a2) and Isabella Novik (a3)

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.

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