The double spherical cap rearrangement of planar sets

Abstract

In the theory of shape optimization, the rearrangements of sets are a key concept, because they allow us to keep some properties of the original set while improving other aspects. This paper is devoted to the proof of an isoperimetric property of the double spherical cap rearrangement of planar sets. In particular, we prove that, under the assumption of disconnection of non-trivial spherical slices, the rearranged set has a lower perimeter than the original one. In the general case, the symmetrized set does not decrease the perimeter, but we show that the “excess” is bounded above by $2{\mathcal{H}}^1(\Gamma )$, where $\Gamma$ denotes the set of radii such that the spherical slice is a non-trivial arc of a circle. Additionally, the higher-dimensional case is briefly discussed; in particular, an explicit counterexample is given, thus explaining why an analogous result cannot hold. The main reason for this is that, in dimension $N=3$ or higher, the union of two spherical caps of equal size does not minimize the $(N-2)$-dimensional measure of the boundary.