The Convex Body Isoperimetric Conjecture in the Plane

TLDR: In this article, a sharp lower bound for the isoperimetric profile of the disk in the plane of a regular polygons has been established for small areas of a planar convex body, where the least perimeter needed to enclose a volume within a polygon is greater than the minimum perimeter needed for enclosing the same volume within any other polygon in Rn.

Abstract: 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. Acknowledgements: This paper is the work of the Williams College NSF “SMALL” 2013, 2014, and 2015 Geometry Groups. We thank our advisor Professor Frank Morgan for his support. We would like to thank the National Science Foundation, Williams College, and the MAA for supporting the “SMALL” REU and our travels to MathFest 2013, MathFest 2014, and MathFest 2015. Page 10 RHIT Undergrad. Math. J., Vol. 18, No. 2