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 and 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 is a circle on , and is a closed curve contained in then .
It is not known whether this result holds for a simple closed convex curve on .
It is known also that this inequality does not hold in the hyperbolic plane.
|Date of creation||2013-03-22 15:31:14|
|Last modified on||2013-03-22 15:31:14|
|Last modified by||PrimeFan (13766)|