This is an announcement for the paper "Mixed integrals and related inequalities" by Vitali Milman and Liran Rotem.
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 inequality, 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.
Archive classification: math.FA math.MG
Mathematics Subject Classification: 52A39, 26B25
Remarks: 30 pages
Submitted from: liranro1@post.tau.ac.il
The paper may be downloaded from the archive by web browser from URL
http://front.math.ucdavis.edu/1210.4346
or