Notes on Schneider’s stability estimates for convex sets

Abstract

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 \({\mathbb{E}^n}\). A unique feature of his approach is to express fundamental geometric quantities in terms of a single function \({\rho:\mathfrak{B} \times \mathfrak{B} \to \mathbb{R}}\) defined on the family of all convex bodies \({\mathfrak{B}}\) in \({\mathbb{E}^n}\). In this paper we show that (the logarithm of) the symmetrized ρ gives rise to a pseudo-metric dD on \({\mathfrak{B}}\) inducing, from our point of view, a finer topology than Banach–Mazur’s dBM. Further, dD induces a metric on the quotient \({\mathfrak{B}/{\rm Dil}^+}\) of \({\mathfrak{B}}\) by the relation of positive dilatation (homothety). Unlike its compact Banach–Mazur counterpart, dD is only “boundedly compact,” in particular, complete and locally compact. The general linear group \({{\rm GL}(\mathbb{E}^n)}\) acts on \({\mathfrak{B}/{\rm Dil}^+}\) by isometries with respect to dD, and the orbit space is naturally identified with the Banach–Mazur compactum \({\mathfrak{B}/{\rm Aff}}\) via the natural projection \({\pi:\mathfrak{B}/{\rm Dil}^+\to\mathfrak{B}/{\rm Aff}}\), where Aff is the affine group of \({\mathbb{E}^n}\). The metric dD has the advantage that many geometric quantities are explicitly computable. We show that dD 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 dBM replaced by dD.