The Minimum Principle for Convex Subequations

Abstract

A subequation on an open subset $X\subset \mathbb R^n$ 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 $\mathcal P$ denote the subequation consisting of those $2$-jets whose Hessian part is semipositive. We introduce a notion of product subequation $F\#\mathcal P$ on $X\times \mathbb R^{m}$ and prove, under suitable hypotheses, that if $F$ is convex and $f(x,y)$ is $F\#\mathcal P$-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.