  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.
