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# DNA inequality

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.

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.

## Mathematics Subject Classification

53A04*no label found*

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