= Submodularity of relative perimeter
{title2=$\operatorname{Per}(E\cap F)+\operatorname{Per}(E\cup F)\le\operatorname{Per}(E)+\operatorname{Per}(F)$}
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.
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