The Minimum Principle for Convex Subequations

Journal article, 2022

A subequation, in the sense of Harvey–Lawson, on an open subset X⊂ Rn 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× Rm and prove, under suitable hypotheses, that if F is convex and f(x, y) is F# P-subharmonic then the marginal function g(x):=infyf(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.