Affine subspaces of curvature functions from closed planar curves

TL;DR

This paper investigates conditions under which a family of curvature functions derived from pairs of real functions can represent closed planar curves, exploring both continuous and discrete cases, and discusses implications for the 4-vertex theorem.

Contribution

It provides new conditions and explicit constructions for curvature functions of closed planar curves and analyzes the discrete analogue, extending classical differential geometry results.

Findings

Identifies equivalent conditions for curvature functions of closed curves.

Constructs explicit examples of such curvature pairs.

Shows limitations in extending the 4-vertex theorem analogues.

Abstract

Given a pair of real functions $(k,f)$, we study the conditions they must satisfy for $k+\lambda f$ to be the curvature in the arc-length of a closed planar curve for all real $\lambda$. Several equivalent conditions are pointed out, certain periodic behaviours are shown as essential and a family of such pairs is explicitely constructed. The discrete counterpart of the problem is also studied. Finally, the characterization obtained is used to show that a sufficient analogue of the 4-vertex theorem cannot be developed.