DNA inequality

Given Γ, a convex simple closed curve (http://planetmath.org/Curve) in the plane, and γ a closed curve contained in Γ, then M(Γ)M(γ) where M is the mean curvatureMathworldPlanetmathPlanetmathPlanetmath 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 Γ and γ 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 Γ is a circle on S2 , and γ is a closed curve contained in Γ then M(Γ)M(γ) .

It is not known whether this result holds for Γ a simple closed convex curve on S2.

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