How to take shortcuts in Euclidean space: making a given set into a short quasi-convex set (Q2909058)

A set \(\Gamma\subset \mathbb{R}^d\) is called \(C-\)quasi-convex if for any \(x,y\in\Gamma\) there is a path \(\gamma\subset\Gamma\) connecting \(x\) and \(y\) such that the length of \(\gamma\) does not exceed \(C|x-y|.\) The main result of the paper is the following one.NEWLINENEWLINETheorem. There exist an absolute constant \(C_1\) and depending on an integer \(d\geq 2\) a constant \(C_2 >1\) such that any set \(K\subset\mathbb{R}^d\) can be covered by a \(C_{2}-\)quasi-convex connected set \(\tilde{\Gamma}\) satisfying the inequality \({\mathbb{H}}^{1}(\tilde{\Gamma})\leq C_{1}{\mathbb{H}}^{1}(\Gamma)\) for any connected \(\Gamma\supset K.\) The symbol \({\mathbb{H}}^1\) stands here for the 1-dimensional Hausdorff measure.NEWLINENEWLINEFor \(d=2\) this result was obtained by \textit{P. W. Jones} [Invent. Math. 102, No. 1, 1--15 (1990; Zbl 0731.30018)].NEWLINENEWLINEThe authors obtain also a certain condition for the independence of the constant \(C_2\) of dimension \(d\).