  We develop a new framework for establishing approximate factorization of
entropy on arbitrary probability spaces, using a geometric notion known as
non-negative sectional curvature. The resulting estimates are equivalent to
entropy subadditivity and generalized Brascamp-Lieb inequalities, and provide a
sharp modified log-Sobolev inequality for the Gibbs sampler of several particle
systems in both continuous and discrete settings. The method allows us to
obtain simple proofs of known results, as well as some new inequalities. We
illustrate this through various applications, including discrete Gaussian free
fields on arbitrary networks, the down-up walk on uniform $n$-sets, the uniform
measure over permutations, and the uniform measure on the unit sphere in
$\R^n$. Our method also yields a simple, coupling-based proof of the celebrated
logarithmic Sobolev inequality for Langevin diffusions in a convex potential,
which is one of the most emblematic applications of the Bakry-\'Emery
criterion.
