  This paper establishes a necessary and sufficient condition for
$L^p$-boundedness of a class of multilinear functionals which includes both the
Brascamp-Lieb inequalities and generalized Radon transforms associated to
algebraic incidence relations. The testing condition involves bounding the
average of an inverse power of certain Jacobian-type quantities along fibers of
associated projections and covers many widely-studied special cases, including
convolution with measures on nondegenerate hypersurfaces or on nondegenerate
curves. The heart of the proof is based on Guth's visibility lemma in one
direction and on a careful analysis of Knapp-type examples in the other.
Various applications are discussed which demonstrate new and subtle interplay
between curvature and transversality and establish nontrivial mixed-norm
$L^p$-improving inequalities in the model case of convolution with affine
hypersurface measure on the paraboloid.
