Shur type comparison theorems for affine curves with application to lattice point estimates

Abstract

If $c, \overline c\colon [a,b]\to \mathbb R^2$ are two convex planar curve parameterized by affine arc length and $A\colon [a,b]\to [0,\infty)$ is the area bounded by the restriction $c\big|_{[a,s]}$ and the segment between $c(a)$ and $c(s)$ with $\overline A$ the corresponding function for $\overline c$, and the affine curvature of $c$ pointwise less than the affine curvature of $\overline c$. Then $A(s)\ge \overline A(s)$. Also for any point of a convex curve we define \emph{adapted affine coordinates} centered at the point and give sharp estimates on the coordinates of the curve in terms of bounds on the curvature. Proving these bounds involves generalizing classical comparison theorems of Strum-Liouville type to higher order and nonhomogenous equations. These estimates allow us to give sharp bounds on the areas of inscribed triangles in terms of affine curvature and the affine distance between the vertices. These inequalities imply upper bounds of the number of lattice points on a convex curve in terms of its affine arc length.