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 d _D on ${\mathfrak{B}}$ inducing, from our point of view, a finer topology than Banach–Mazur’s d _BM . Further, d _D induces a metric on the quotient ${\mathfrak{B}/{\rm Dil}^+}$ of ${\mathfrak{B}}$ 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 ${{\rm GL}(\mathbb{E}^n)}$ acts on ${\mathfrak{B}/{\rm Dil}^+}$ by isometries with respect to d _D , 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 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 .