On the unfolding of simple closed curves

TL;DR

This paper proves that any simple closed curve in the plane can be continuously deformed into a convex shape without decreasing distances between points, extending polygon results to smooth curves through approximation and generalized tools.

Contribution

It introduces a continuous deformation process for simple closed curves, generalizes key geometric tools, and advances towards a complete continuous proof of convexification.

Findings

Every rectifiable simple closed curve can be convexified via a distance-preserving deformation.

The approach extends polygon-based results to smooth curves through approximation.

Generalizations of Farkas Lemma and Maxwell-Cremona Theorem are developed for measures on the plane.

Abstract

I show that every rectifiable simple closed curve in the plane can be continuously deformed into a convex curve in a motion which preserves arc length and does not decrease the Euclidean distance between any pair of points on the curve. This result is obtained by approximating the curve with polygons and invoking the result of Connelly, Demaine, and Rote that such a motion exists for polygons. I also formulate a generalization of their program, thereby making steps toward a fully continuous proof of the result. To facilitate this, I generalize two of the primary tools used in their program: the Farkas Lemma of linear programming to Banach spaces and the Maxwell-Cremona Theorem of rigidity theory to apply to stresses represented by measures on the plane.