DNA inequality

Given $\mathrm{\Gamma }$, a convex simple closed curve (http://planetmath.org/Curve) in the plane, and $\gamma$ a closed curve contained in $\mathrm{\Gamma }$, then $M(\mathrm{\Gamma })\le M(\gamma )$ where $M$ is the mean curvature function.

This was a conjecture due to S. Tabachnikov and was proved by Lagarias and Richardson of Bell Labs. The idea of the proof was to show that there was a way you could reduce a curve to the boundary of its convex hull so that if it holds for the boundary of the convex hull, then it holds for the curve itself.

Conjecture : Equality holds iff $\mathrm{\Gamma }$ and $\gamma$ coincide.

It’s amazing how many questions are still open in the Elementary Differential Geometry of curves and surfaces. Questions like this often serve as a great research opportunity for undergraduates. It is also interesting to see if you could extend this result to curves on surfaces:

Theorem : If $\mathrm{\Gamma }$ is a circle on $S^2$ , and $\gamma$ is a closed curve contained in $\mathrm{\Gamma }$ then $M(\mathrm{\Gamma })\le M(\gamma )$ .

It is not known whether this result holds for $\mathrm{\Gamma }$ a simple closed convex curve on $S^2$.

It is known also that this inequality does not hold in the hyperbolic plane.