= Comparison of perimeter minimizers with ordered forcing
{title2=$g<h\text{ a.e.}\quad\Longrightarrow\quad|E_g\setminus E_h|=0$}
Suppose $E_g$ minimizes $\operatorname{Per}(E)-\int_Eg$ and $E_h$ minimizes $\operatorname{Per}(E)-\int_Eh$. Compare them with their intersection and union. Adding their optimality inequalities and using <submodularity of relative perimeter> gives $\int_{E_g\setminus E_h}(h-g)\le0$. Strict ordering forces that set to be null. Applied to $g=(f-t)/\alpha$, this yields nested <ROF denoising> <superlevel sets>.
Back to article page
