Isotropic constants and regular polytopes

We discuss first-order optimality conditions for the isotropic constant and combine them with RS-movements to obtain structural information about polytopal maximizers. Strengthening a result by Rademacher, it is shown that a polytopal local maximizer with a simplicial vertex must be a simplex. A similar statement is shown for a centrally symmetric local maximizer with a simplicial vertex: it has to be a cross-polytope. Moreover, we show that a zonotope that maximizes the isotropic constant and that has a cubical zone must be a cube. Finally, we consider the class of zonotopes with at most n+1 generators and determine the extremals in this class.