The perimeter inequality under Steiner symmetrization: cases of equality.

*(English)*Zbl 1087.28003The 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.

Reviewer: Salvador Gomis (Alicante)

##### 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 |