  If $f,g:\mathbb{R}^n\longrightarrow\mathbb{R}_{\geq0}$ are non-negative
measurable functions, then the Pr\'ekopa-Leindler inequality asserts that the
integral of the Asplund sum (provided that it is measurable) is greater or
equal than the $0$-mean of the integrals of $f$ and $g$. In this paper we prove
that under the sole assumption that $f$ and $g$ have a common projection onto a
hyperplane, the Pr\'ekopa-Leindler inequality admits a linear refinement.
Moreover, the same inequality can be obtained when assuming that both
projections (not necessarily equal as functions) have the same integral. An
analogous approach may be also carried out for the so-called
Borell-Brascamp-Lieb inequality.
