  We introduce a new operation between nonnegative integrable functions on
$\mathbb{R} ^n$, that we call geometric combination; it is obtained via a mass
transportation approach, playing with inverse distribution functions. The main
feature of this operation is that the Lebesgue integral of the geometric
combination equals the geometric mean of the two separate integrals; as a
natural consequence, we derive a new functional inequality of
Pr\'ekopa-Leindler type. When applied to the characteristic functions of two
measurable sets, their geometric combination provides a set whose volume equals
the geometric mean of the two separate volumes. In the framework of convex
bodies, by comparing the geometric combination with the $0$-sum, we get an
alternative proof of the log-Brunn-Minkowski inequality for unconditional
convex bodies and for convex bodies with $n$ symmetries.
