Perimeter quasiminimizer ($\mathrm{Per}(E;U)\le \mathrm{Per}(F;U)+\Lambda |E△F|$)

Here the sets of finite perimeter $E,F$ differ only compactly inside $U$. A minimizing set with bounded volume forcing satisfies this inequality. Interior regularity theorems give graph patches outside a singular set negligible for surface Hausdorff measure; bounded forcing allows the graph estimates used in the noncontact of ROF level boundaries.