Weighted geometric means of convex bodies

Vitali Milman, Liran Rotem

Research output: Chapter in Book/Report/Conference proceedingChapterpeer-review

Abstract

Given two convex bodies K and T and a number 0 < λ < 1, we present two constructions for the geometric mean of K and T with weights 1 − λ and λ. If K and T are the unit balls of two norms, one may think of the weighted geometric means as the unit balls of a “geometric interpolation” between the two normed spaces. Our first construction is explicit, but lacks the so called “duality prop-erty” – a natural property one expects from the geometric mean. The second construction has this property, but is based on the choice of an ultrafilter. Both definitions extend previous constructions by the authors for the power Kλ, and the results proved here also prove some previously announced results.

Original languageEnglish
Title of host publicationContemporary Mathematics
PublisherAmerican Mathematical Society
Pages233-249
Number of pages17
DOIs
StatePublished - 2019

Publication series

NameContemporary Mathematics
Volume733
ISSN (Print)0271-4132
ISSN (Electronic)1098-3627

Keywords

  • Convex geometry
  • Geometric mean
  • Positive-definite matrices
  • Power

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