Archive for July 2009

Sectors with Density in Granada

My undergraduate research Geometry Group and I have been having a great summer here in Granada Spain. We’ve been considering planar sectors of angle 0<\theta<\infty with density r^p (p>0) and the isoperimetric problem: to enclose given weighted area with least weighted perimeter. We’ve proved that there are angles 0<\theta_1<\theta_2 \leq\pi such that the minimizer is:

1. for 0<\theta < \theta_1, a circular arc about the origin;

2. for \theta_1<\theta < \theta_2, an unduloid (half-period of a periodic curve normal to both edges of the sector);

3. for \theta_2<\theta, a semicircle through the origin.

We have lots of evidence that \theta_1=\pi/\sqrt{p+1} and \theta_2=\pi(p+2)/(2p+2), but we have not been able to prove it. Can you help us? Check out our arXiv post.