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Let $E$ have finite perimeter in $\mathbb R^n$. Consider the minimization of

$$P(B)-\int_{\mathbb R^n\setminus L}|D\chi_E|$$

among the sets $B$ of finite perimeter differing from $E$ only inside $L$ (the negative piece of the minimized quantity is not influent, since it does not depend on $B$, but it's there to stress the idea that we care just about the perimeter inside $L$). Let $\bar E$ be the minimizer, and that the minimum is not zero. Suppose also that $L$ has finite perimeter.

I'm looking for a geometrically inspired way for defining a function $\Psi$ which sends a piece (I would like to ask "all" but it might be unfeasible) of $\mathcal FL\cap E$ to $\mathcal F\bar E\cap L$ bijectively, is $H^{n-1}$-measurable and decreases (more precisely, does not increase) $(n-1)$-dimensional area.

The idea should be that $\Psi$ be some kind of projection.

In the case that $L$ is a ball and $E$ is a thin rectangle around its diameter it is clear that usual projections don't quite work.

On the other hand, such a map surely exists, by the consideration that the area of the target is smaller than the one of the domain, and both of them are rectifiable.

My question is if there is a famous geometric/constructive way of doing it, perhaps using some minimization principle, a flow, etc., or under more restrictive hypotheses on $L$.

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It seems this question is not yet fully specified. What is $P$? What is $D$? What type of objects are $E$ and $L$? –  Joseph O'Rourke Feb 2 '12 at 14:44
1  
Both $P$ and $D$ are standard notation in this context. $P$ is the perimeter and $D$ is the distributional derivative. $E$ and $L$ are usually Borel sets (with possibly some extra regularity if needed). Also, $\mathcal F$ denotes the reduced boundary and $H^{n-1}$ is the $n-1$-dimensional Hausdorff measure. –  Tapio Rajala Feb 2 '12 at 15:06
    
dear Joseph, sorry for the sloppy notations. They are meant as Tapio suggested. –  Mircea Feb 2 '12 at 15:09
    
Thanks, Tapio & Mircea. –  Joseph O'Rourke Feb 2 '12 at 15:18
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