The Minimum Principle for Convex Subequations

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 $F\mathrm{\#}\mathcal{P}$ on $X\times \mathbb R^{m}$andprove,undersuitablehypotheses,thatif$F$isconvexand$f(x,y)$is$F#\mathcal P$-subharmonicthenthemarginalfunction$$ g(x):= \inf_y f(x,y)$$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.