On local maximizers of the isotropic constant

The well-known slicing problem in convex geometry is equivalent to the question of whether the isotropic constant L_K of an arbitrary convex body K is bounded from above by a universal constant. The isotropic constant conjecture, which was recently proven by Klartag and Lehec, asserts that such a constant exists. According to a strong version of the isotropic constant conjecture, the isotropic constant is maximized by the simplex in every dimension n ≥ 1. In this thesis, we study local maximizers of the function K ↦ L_K in a given dimension n and establish necessary conditions for such maximizers. The main part of the thesis consists of three chapters. In Chapter 1, we study an abstract notion of small (i.e., infinitesimal) perturbations of polytopes, which is motivated by our overarching goal of finding first-order necessary conditions for local maximizers of the isotropic constant. Recovering a notion of weakly differentiable convex-body-valued maps due to Pflug and Weisshaupt, it is shown that such a small perturbation can be represented by a signed measure on the boundary of the perturbed body K. We consider the question which signed measures can be realized as small perturbations of a given convex body and give a full solution to this question in the case where K is a polytope. In Chapter 2, we apply the first-order necessary conditions obtained in Chapter 1 to identify structural properties of polytopal local maximizers of K ↦ L_K. Strengthening a result by Rademacher, it is shown that a polytopal local maximizer of the isotropic constant with a simplicial vertex has to be a simplex. Similar results are shown for centrally symmetric maximizers with a simplicial vertex and zonotopal maximizers in the class of centrally symmetric convex bodies; such polytopes have to be a cross-polytope and a parallelepiped, respectively. Finally, the class of zonotopes with at most n+1 generators is considered and the extremizers of K ↦ L_K in this class are determined. In Chapter 3, we discuss necessary conditions for local maximizers of K ↦ L_K that are related to notions of decomposability. A result by Campi, Colesanti and Gronchi concerning RS-decomposability is extended to a larger class of shadow systems. Moreover, we show that the polar body K° of a local maximizer K of the isotropic constant cannot have too many Minkowski summands; more precisely, its dimension of decomposability is at most ½(n²+3n). We discuss the polytopal case, which turns out to have connections to affine rigidity theory, and show how the bound on the dimension of decomposability of K° can be improved if K is assumed to have certain symmetries. The thesis ends with a chapter containing "Concluding remarks" on some aspects of Chapters 1, 2 and 3.