  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. We consider a
centro-affine analog of L\"uk\H{o}'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.
