Neighborliness of the symmetric moment curve

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We consider the convex hull Bₖ of the symmetric moment curve U(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/2pi Z. The curve U(t) is locally neighborly: as long as t₁,..., tₖ lie in an open arc of S of a certain length phiₖ>0, the convex hull of the points U(t₁),..., U(tₖ) is a face of Bₖ. We characterize the maximum possible length phiₖ, proving, in particular, that phiₖ > pi/2 for all k and that the limit of phiₖ is pi/2 as k grows. This allows us to construct centrally symmetric polytopes with a record number of faces.