Geometry of log-concave functions and measures

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Abstract

We present a view of log-concave measures, which enables one to build an isomorphic theory for high dimensional log-concave measures, analogous to the corresponding theory for convex bodies. Concepts such as duality and the Minkowski sum are described for log-concave functions. In this context, we interpret the Brunn-Minkowski and the Blaschke-Santaló inequalities and prove the two corresponding reverse inequalities. We also prove an analog of Milman's quotient of subspace theorem, and present a functional version of the Urysohn inequality.

Original languageEnglish
Pages (from-to)169-182
Number of pages14
JournalGeometriae Dedicata
Volume112
Issue number1
DOIs
StatePublished - Apr 2005

Keywords

  • Geometric inequalities
  • Log-concave functions
  • Log-concave measures
  • Reverse Brunn-Minkowski
  • Reverse Santalò

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