## 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.

Right now we know that for density $r^p$ the circular arc minimizes up until the π/(p+1) sector. We could improve that to the π/(p/2+1) sector if we could show that in the plane with regular area and perimeter density $r^q$ for $0 circles about the origin were minimizers. Indeed, the entire conjecture has an equivalent statement in terms of the plane with different area and perimeter densities, but this is a particularly nice case. Furthermore the line p/2+1 is the tangent line at p=0 to √(p+1) so this is the best possible linear (in the denominator) estimate we could get.