# 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.

Title | DNA inequality |
---|---|

Canonical name | DNAInequality |

Date of creation | 2013-03-22 15:31:14 |

Last modified on | 2013-03-22 15:31:14 |

Owner | PrimeFan (13766) |

Last modified by | PrimeFan (13766) |

Numerical id | 12 |

Author | PrimeFan (13766) |

Entry type | Theorem |

Classification | msc 53A04 |