  Given any (forward) Brascamp--Lieb inequality on euclidean space, a famous
theorem of Lieb guarantees that gaussian near-maximizers always exist.
Recently, Barthe and Wolff used mass transportation techniques to establish a
counterpart to Lieb's theorem for all non-degenerate cases of the inverse
Brascamp--Lieb inequality. Here we build on work of Chen--Dafnis--Paouris and
employ heat-flow techniques to understand the inverse Brascamp--Lieb inequality
for certain regularized input functions, in particular extending the
Barthe--Wolff theorem to such a setting. Inspiration arose from work of
Bennett, Carbery, Christ and Tao for the forward inequality, as well as Wolff's
surprising observation that the gaussian saturation property for the inverse
Brascamp--Lieb inequality implies the gaussian saturation property for both the
forward Brascamp--Lieb inequality and Barthe's reverse Brascamp--Lieb
inequality.
