Geometric measures in the dual Brunn–Minkowski theory and their associated Minkowski problems

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Abstract

A longstanding question in the dual Brunn–Minkowski theory is “What are the dual analogues of Federer’s curvature measures for convex bodies?” The answer to this is provided. This leads naturally to dual versions of Minkowski-type problems: What are necessary and sufficient conditions for a Borel measure to be a dual curvature measure of a convex body? Sufficient conditions, involving measure concentration, are established for the existence of solutions to these problems.

Original languageEnglish (US)
Pages (from-to)325-388
Number of pages64
JournalActa Mathematica
Volume216
Issue number2
DOIs
StatePublished - Jun 1 2016

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Keywords

  • Alexandrov problem
  • cone-volume measure
  • dual Brunn–Minkowski theory
  • dual curvature measure
  • integral curvature
  • L-Minkowski problem
  • logarithmic Minkowski problem
  • Minkowski problem
  • surface area measure

ASJC Scopus subject areas

  • Mathematics(all)

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