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 Lükő’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.
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Tabachnikov, S. Variations on R. Schwartz’s Inequality for the Schwarzian Derivative. Discrete Comput Geom 46, 724–742 (2011). https://doi.org/10.1007/s00454-011-9371-7
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DOI: https://doi.org/10.1007/s00454-011-9371-7