  We initiate a systematic study of intrinsic dimensional versions of classical
functional inequalities which capture refined properties of the underlying
objects. We focus on model spaces: Euclidean space, Hamming cube, and manifolds
of constant curvature. In the latter settings, our intrinsic dimensional
functional inequalities improve on a series of known results and lead to new
Hamilton-type matrix inequalities. Our proofs rely on scaling, tensorization,
and stochastic methods.
