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Edges of the Barvinok–Novik Orbitope

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Abstract

Here we study the kth symmetric trigonometric moment curve and its convex hull, the Barvinok–Novik orbitope. In 2008, Barvinok and Novik introduced these objects and showed that there is some threshold so that for two points on \(\mathbb {S}^{1}\) with arclength below this threshold the line segment between their lifts to the curve forms an edge on the Barvinok–Novik orbitope, and for points with arclength above this threshold their lifts do not form an edge. They also gave a lower bound for this threshold and conjectured that this bound is tight. Results of Smilansky prove tightness for k=2. Here we prove this conjecture for all k.

References

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    Smilansky, Z.: Convex hulls of generalized moment curves. Isr. J. Math. 52, 115–128 (1985)

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Author information

Correspondence to Cynthia Vinzant.

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Vinzant, C. Edges of the Barvinok–Novik Orbitope. Discrete Comput Geom 46, 479–487 (2011). https://doi.org/10.1007/s00454-011-9351-y

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Keywords

  • Moment curve
  • Toeplitz operator
  • Orbitope
  • Convex hull of a curve