Given $\Gamma$, a convex simple closed curve in the plane, and $\gamma$ a closed curve contained in $\Gamma$, then $M(\Gamma)\leq M(\gamma)$ 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 $\Gamma$ 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 $\Gamma$ is a circle on $S^{2}$ , and $\gamma$ is a closed curve contained in $\Gamma$ then $M(\Gamma)\leq M(\gamma)$ .

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

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