Minimum Principle for Convex Subequations

A subequation, in the sense of Harvey-Lawson, on an open subset X subset of R-n is a subset F of the space of 2-jets on X with certain properties. A smooth function is said to be F-subharmonic if all of its 2-jets lie in F, and using the viscosity technique one can extend the notion of F-subharmonicity to any upper-semicontinuous function. Let P denote the subequation consisting of those 2-jets whose Hessian part is semipositive. We introduce a notion of product subequation F#P on X x R-m and prove, under suitable hypotheses, that if F is convex and f (x, y) is F#P-subharmonic then the marginal function g(x) := inf y f (x, y) is F-subharmonic. This generalises the classical statement that the marginal function of a convex function is again convex. We also prove a complex version of this result that generalises the Kiselman minimum principle for the marginal function of a plurisubharmonic function.