  This paper considers the problem of $L^p$-estimates for a certain multilinear
functional involving integration against a kernel with the structure of a
determinant. Examples of such objects are ubiquitous in the study of Fourier
restriction and geometric averaging operators. It is shown that, under very
general circumstances, the boundedness of such functionals is equivalent to a
geometric inequality for measures which has recently appeared in work by D.
Oberlin (Math Proc. Cambridge. Philos. Soc., 129, 2000) and Bak, Oberlin, and
Seeger (J. Aust. Math. Soc., 85, 2008).
