Schur type comparison theorems for affine curves

Abstract

Let $c, \overline{c}:[a,b]\rightarrow {\mathbb R}^2$ are two convex planar curve parameterized by affine arc length and A is the area bounded by the image of c and its endpoints and $\overline{A}$ the corresponding area for $\overline{c}$. If the affine curvatures are related by ${\varkappa }(s) \le \overline{{\varkappa }}(s)$, then $A \ge \overline{A}$. If either ${\varkappa }$ or $\overline{{\varkappa }}$ is $C^1$ then equality holds if and only if $\overline{c}$ is the image of c under a special affine motion. Also for any point of a convex curve we define 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 upper bounds on the areas of triangles with whose vertices are on a curve in terms of the curvature and a affine length between the points.