Notes on Schneider’s Stability Estimates for Convex Sets in Minkowski Space
نویسنده
چکیده
In 2009 [Schneider 1] obtained stability estimates in terms of the BanachMazur distance for several geometric inequalities for convex bodies in an ndimensional Minkowski 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 set of all convex bodies B in En. In this paper we show that (the logarithm of) the symmetrized ρ gives rise to a pseudo-metric dD on B inducing a finer topology than Banach-Mazur’s dBM . Further, dD induces a metric on the quotient B/Dil of 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 GL(En) acts on B/Dil by isometries with respect to dD, and the orbits 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 En. The metric dD has the advantage that many geometric quantities are explicitly computable. We will show that dD provides a simpler and more fitting environment for the study of stability; in particular, all the estimates of [Schneider 1] turn out to be valid with dBM replaced by dD. 1 A Positive-Dilatation Invariant Pseudo-Metric Let E be a Minkowski space of dimension n, and denote by B the set of all convex bodies in E. We emphasize here that we only consider convex bodies that have non-empty interior in E, that is, all the members of B have dimension n. Let Aff = Aff (E) denote the affine group, the Lie group of affine transformations of E. (For brevity E will be suppressed from the notation.) It can be written as the semi-direct product Aff = T o GL, where T ∼= E is the (additive) group of
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