Optimal regularity for quasiminimal sets of codimension one in ${\mathbb{R}}^2$ and ${\mathbb{R}}^3$

Abstract

Quasiminimal sets are sets for which a pertubation can decrease the area but only in a controlled manner. We prove that in dimensions $2$ and $3$, such sets separate a locally finite family of local John domains. Reciprocally, we show that this property is a sufficient for quasiminimality. In addition, we show that quasiminimal sets locally separate the space in two components, except at isolated points in ${\mathbb{R}}^2$ or out a of subset of dimension strictly less than $N-1$ in ${\mathbb{R}}^N$.