A 4-point theorem: still another variation on an old theme

TL;DR

This paper revisits Graustein's theorem on the four points of average curvature in plane ovals, providing a new proof via wave propagation and extending the result to spherical and hyperbolic geometries.

Contribution

It introduces a novel proof method using wave propagation and generalizes the theorem to non-Euclidean geometries.

Findings

The average curvature points are at least four for plane ovals.

The theorem extends to spherical geometries.

The theorem applies to horocyclically convex curves in hyperbolic geometry.

Abstract

An old theorem, due to Graustein, asserts that the average curvature of a plane oval is attained at least at four points. We present a proof by way of wave propagation and extend this result to the spherical and hyperbolic geometries - in the latter case, to horocyclically convex curves only.