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
