  Mixed volumes, which are the polarization of volume with respect to the
Minkowski addition, are fundamental objects in convexity. In this note we
announce the construction of mixed integrals, which are functional analogs of
mixed volumes. We build a natural addition operation + on the class of
quasi-concave functions, such that every class of \alpha-concave functions is
closed under +. We then define the mixed integrals, which are the polarization
of the integral with respect to +.
  We proceed to discuss the extension of various classic 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 \alpha-concave functions, we state a generalization of
the Alexandrov inequalities in their more familiar form.
