Sullivan's lamination of a planar region (Q1071313)

The following theorem, which is close to the Sullivan's lamination theorem, is presented with a very simple proof based on the nonstandard analysis transcription of the original Sullivan's idea: Let R be a bounded open set in \({\mathbb{R}}^ 2\), B be the boundary of R; then for every \(p\in R\), there exists a circle K in \({\mathbb{R}}^ 2\) such that the open region bounded by K is included in R and \(p\in conv(K\cap B)\). (The possibility of direct generalization to the case of \({\mathbb{R}}^ n\) is mentioned.)