  We provide functional analogues of the classical geometric inequality of
Rogers and Shephard on products of volumes of sections and projections. As a
consequence we recover (and obtain some new) functional versions of
Rogers-Shephard type inequalities as well as some generalizations of the
geometric Rogers-Shephard inequality in the case where the subspaces intersect.
These generalizations can be regarded as sharp local reverse Loomis-Whitney
inequalities. We also obtain a sharp local Loomis-Whitney inequality.
