Notes on Schneider's stability estimates for convex sets

In 2009 Schneider obtained stability estimates in terms of the Banach-Mazur distance for several geometric inequalities for convex bodies in an n-dimensional normed space Eⁿ. A unique feature of his approach is to express fundamental geometric quantities in terms of a single function ρ: B×B → ℝ defined on the family of all convex bodies B in E_n. In this paper we show that (the logarithm of) the symmetrized ρ gives rise to a pseudo-metric d_D on B inducing, from our point of view, a finer topology than Banach-Mazur's d_BM. Further, d_D induces a metric on the quotient B/Dil⁺ of B by the relation of positive dilatation (homothety). Unlike its compact Banach-Mazur counterpart, d_D is only "boundedly compact," in particular, complete and locally compact. The general linear group GL(Eⁿ) acts on B/Dil+ by isometries with respect to dD, and the orbit space is naturally identified with the Banach-Mazur compactum B/Aff via the natural projection π: B/Dil⁺ → B/Aff, where Aff is the affine group of E_n. Themetric d_D has the advantage that many geometric quantities are explicitly computable. We show that d_D provides a simpler and more fitting environment for the study of stability; in particular, all the estimates of Schneider turn out to be valid with d_BM replaced by d_D.