The length of a set in the sphere whose polynomial hull contains the origin
Definition. (a) A set \(X<\mathbb{R}^ k\) is 1-rectifiable if it is the image of a bounded subset \(U\subset\mathbb{R}\) under a Lipschitz continuous mapping \(f:U\to\mathbb{R}^ k\). (b) \(X\) is \(({\mathcal H}^ 1,1)\)-rectifiable if \({\mathcal H}^ 1(X)<\infty\) and \({\mathcal H}^ 1\)-almost all of \(X\) can be covered by a countable union of 1-rectifiable sets. \({\mathcal H}^ 1\) denotes here a 1-dimensional Hausdorff measure. The main result of the paper is Theorem. If \(X\) is a compact \(({\mathcal H}^ 1,1)\)-rectifiable subset of the unit sphere \(S\subset\mathbb{C}^ n\) such that the origin \(0\in\mathbb{C}^ n\) belongs to the polynomial hull \(\widehat X\), then \({\mathcal H}^ 1(X)\geq 2\pi\).
scientific article; zbMATH DE number 78086
scientific article