Variations on R. Schwartz's Inequality for the Schwarzian Derivative

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Abstract

R. Schwartz's inequality provides an upper bound for the Schwarzian derivative of a parameterization of a circle in the complex plane and on the potential of Hill's equation with coexisting periodic solutions. We prove a discrete version of this inequality and obtain a version of the planar Blaschke-Santalo inequality for not necessarily convex polygons. In the proof, we use some formulas from the theory of frieze patterns. We consider a centro-affine analog of Lüko{double acute}'s inequality for the average squared length of a chord subtending a fixed arc length of a curve-the role of the squared length played by the area-and prove that the central ellipses are local minima of the respective functionals on the space of star-shaped centrally symmetric curves. We conjecture that the central ellipses are global minima. In an appendix, we relate the Blaschke-Santalo and Mahler inequalities with the asymptotic dynamics of outer billiards at infinity.

Original languageEnglish (US)
Pages (from-to)724-742
Number of pages19
JournalDiscrete and Computational Geometry
Volume46
Issue number4
DOIs
StatePublished - Dec 1 2011

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All Science Journal Classification (ASJC) codes

  • Theoretical Computer Science
  • Geometry and Topology
  • Discrete Mathematics and Combinatorics
  • Computational Theory and Mathematics

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