$\alpha$-concave functions and a functional extension of mixed volumes

Abstract

Mixed volumes, which are the polarization of volume with respect tothe Minkowski addition, are fundamental objects in convexity. In thisnote we announce the construction of mixed integrals, which are functionalanalogs of mixed volumes. We build a natural addition operation $\oplus$on the class of quasi-concave functions, such that every class of$\alpha$-concave functions is closed under $\oplus$. We then definethe mixed integrals, which are the polarization of the integral withrespect to $\oplus$. We proceed to discuss the extension of various classic inequalitiesto the functional setting. For general quasi-concave functions, thisis done by restating those results in the language of rearrangementinequalities. Restricting ourselves to $\alpha$-concave functions,we state a generalization of the Alexandrov inequalities in theirmore familiar form.

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