Submodularity of relative perimeter ($\mathrm{Per}(E\cap F)+\mathrm{Per}(E\cup F)\le \mathrm{Per}(E)+\mathrm{Per}(F)$)

The BV lattice inequality $|D\min (v,w)|+|D\max (v,w)|\le |Dv|+|Dw|$ applied to two indicator functions gives the perimeter inequality. For smooth approximations, minima and maxima partition the two gradients; sequential lower semicontinuity passes the estimate to BV. It prevents a crossing of minimizers with strictly ordered set forcing.

Suppose $E_g$ minimizes $\mathrm{Per}(E)-{\int }_{E}g$ and $E_h$ minimizes $\mathrm{Per}(E)-{\int }_{E}h$. Compare them with their intersection and union. Adding their optimality inequalities and using submodularity of relative perimeter gives ${\int }_{E_g\setminus E_h}(h-g)\le 0$. Strict ordering forces that set to be null. Applied to $g=(f-t)/\alpha$, this yields nested ROF denoising superlevel sets.