Isoperimetry in the Plane with Density e-1/r

We study the isoperimetric problem in the plane with weighting or density e-1/r. The isoperimetric problem seeks to enclose prescribed weighted area with minimum weighted perimeter. For density e-1/r, isoperimetric curves are conjectured to pass through the origin. We provide numerical and theoretical evidence that such curves have an angle at the origin approaching 1 radian from above as area approaches zero and provide further estimates

Surface-area-minimizing n-hedral Tiles

We provide a list of conjectured surface-area-minimizing n-hedral tiles of space for n from 4 to 14, previously known only for n equal to 5 and 6. We find the optimal orientation-preserving tetrahedral tile (n=4), and we give a nice new proof for the optimal 5-hedron (a triangular prism)

The Convex Body Isoperimetric Conjecture in the Plane

The Convex Body Isoperimetric Conjecture states that the least perimeter needed to enclose a volume within a ball is greater than the least perimeter needed to enclose the same volume within any other convex body of the same volume in Rn. We focus on the conjecture in the plane and prove a new sharp lower bound for the isoperimetric profile of the disk in this case. We prove the conjecture in the case of regular polygons, and show that in a general planar convex body the conjecture holds for small areas