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

2024/09/19 by Tabachnikov, Serge
#Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2409.12609

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.

Related

• The Four Vertex Theorem and its Converse
• On Curves and Surfaces of Constant Width
• Some remarks on the correspondence between elliptic curves and four points in the Riemann sphere
• New four-vertex type theorems for spherical polygons
• Four-Manifolds, Curvature Bounds, and Convex Geometry