Notes on Schneider's stability estimates for convex sets

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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 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 En. In this paper we show that (the logarithm of) the symmetrized ρ gives rise to a pseudo-metric dD on B inducing, from our point of view, 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 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 En. Themetric 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.

Original languageEnglish (US)
Pages (from-to)585-598
Number of pages14
JournalJournal of Geometry
Volume104
Issue number3
DOIs
StatePublished - Dec 2013

All Science Journal Classification (ASJC) codes

  • Geometry and Topology

Keywords

  • Banach-Mazur distance
  • Convex body
  • dilatation
  • stability

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