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) |
---|---|
Pages (from-to) | 724-742 |
Number of pages | 19 |
Journal | Discrete and Computational Geometry |
Volume | 46 |
Issue number | 4 |
DOIs | |
State | Published - Dec 2011 |
All Science Journal Classification (ASJC) codes
- Theoretical Computer Science
- Geometry and Topology
- Discrete Mathematics and Combinatorics
- Computational Theory and Mathematics