Computing the Isotropic Ratio Landscape on Polygons

Abstract

Mathematicians have conjectured that the simplexes are the polytopes that maximize the isotropic ratio among all convex bodies in Rn (the Strong Slicing Conjecture). This has been established only in R2, where the triangle is the unique maximizer. In this paper we use both theoretical and computational approaches to derive explicit formulas for the first and second derivatives of the isotropic ratio of a polytope in R2 in terms of its vertices, with the aim of providing a foundation for attacking the conjecture in higher dimensions.