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The perimeter inequality under Steiner symmetrization: cases of equality. (English) Zbl 1087.28003
The main property of perimeter in connection with Steiner symmetrization is that if \(E\) is any set of finite perimeter \(P(E)\) in \(\mathbb{R}^n\), \(n\geq 2\), and \(H\) is any hyperplane, then also its Steiner symmetral \(E^S\) about \(H\) is of finite perimeter and \[ P(E^S) \leq P(E). \] This paper presents a characterization of the sets whose perimeter is preserved under this symmetrization. It is possible to assume without loss of generality that \(H= \{ (x',0): x'\in \mathbb{R}^{n-1} \}\). Let \(\Omega\) denote an open set in \(\mathbb{R}^{n-1}\). The authors find the following minimal assumptions to ensure the equivalence (up to translation) between sets \(E\) and \(E^S\) such that \(P(E^S)=P(E)\): 1) \(\partial ^* E^S\) cannot have flat parts along the \(y\)-axis in \(\Omega \times \mathbb{R}\) with outer \((n-1)\)-dimensional measure strictly positive. (\(\partial ^*\) denotes the reduced boundary operator). 2) No (too large) subset of \(E^S\cap(\Omega \times\mathbb{R})\) shrinks along the \(y\)-axis enough to be contained in \(\Omega \times \{0\}\). The authors also provide a local symmetry result for \(E\) on any cylinder parallel to the \(y\)-axis having the form \(\Omega \times \mathbb{R}\). The proofs involve quite subtle matters requiring delicate tools from geometric measure theory.

MSC:
28A75 Length, area, volume, other geometric measure theory
26B15 Integration of real functions of several variables: length, area, volume
49Q15 Geometric measure and integration theory, integral and normal currents in optimization
52A38 Length, area, volume and convex sets (aspects of convex geometry)
52A40 Inequalities and extremum problems involving convexity in convex geometry
26D20 Other analytical inequalities
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