### 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 language | English (US) |
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Pages (from-to) | 724-742 |

Number of pages | 19 |

Journal | Discrete and Computational Geometry |

Volume | 46 |

Issue number | 4 |

DOIs | |

State | Published - 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