Mixed integrals and related inequalities

Vitali Milman, Liran Rotem

Research output: Contribution to journalArticlepeer-review

15 Scopus citations

Abstract

In this paper we define an addition operation on the class of quasi-concave functions. While the new operation is similar to the well-known sup-convolution, it has the property that it polarizes the Lebesgue integral. This allows us to define mixed integrals, which are the functional analogs of the classic mixed volumes. We extend various classic inequalities, such as the Brunn-Minkowski and the Alexandrov-Fenchel inequalities, to the functional setting. For general quasi-concave functions, this is done by restating those results in the language of rearrangement inequalities. Restricting ourselves to log-concave functions, we prove generalizations of the Alexandrov inequalities in a more familiar form.

Original languageEnglish (US)
Pages (from-to)570-604
Number of pages35
JournalJournal of Functional Analysis
Volume264
Issue number2
DOIs
StatePublished - Jan 15 2013

Bibliographical note

Funding Information:
* Corresponding author. E-mail addresses: milman@post.tau.ac.il (V. Milman), liranro1@post.tau.ac.il (L. Rotem). 1 Partially supported by the Minkowski Center at the University of Tel Aviv, by ISF grant 387/09 and by BSF grant 2006079.

Keywords

  • Alexandrov-Fenchel
  • Brunn-Minkowski
  • Log-concavity
  • Mixed integrals
  • Mixed volumes
  • Quasi-concavity
  • Rescaling

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