  We prove a new family of inequalities, which compare the integral of a
geometric convolution of non-negative functions with the integrals of the
original functions. For classical inf-convolution, this type of inequality is
called the Pr\'{e}kopa-Leindler inequality, which, restricted to indicators of
convex bodies, gives the classical Brunn-Minkowski inequality. The convolution
we consider is a different one, which arises from the study of the polarity
transform for functions. While inf-convolution arises as the pull back of usual
addition of convex functions under the Legendre transform, our geometric
inf-convolution arises as the pull back of the second order reversing transform
on geometric convex functions (called either polarity transform or
$A$-transform). These are, up to linear terms, the only order reversing
isomorphisms on the class of geometric convex functions. We prove that the
integral of this new geometric convolution of two functions is bounded from
below by the harmonic average of the individual integrals. Our inequality
implies the Brunn-Minkowski inequality, as well as some other, new,
inequalities for volumes of bodies. Our inequalities are intimately connected
with Busemann's convexity theorem, a new variant of which we prove for 1-convex
hulls and log-concave densities.
