On one-dimensional continua uniformly approximating planar sets (Q2509085)

The authors consider the two following problems: {\parindent=7mm \begin{itemize}\item[P1:] Let \(M\), \(A\) be compact sets of the \(n\)-dimensional Euclidean space, let \(A\) be connected with prescribed one-dimensional Hausdorff measure less or equal to a fixed positive real number \(l\). Minimize the functional \(F_M(A):= \max_M\,\text{dist}(y,A)\) over all sets \(A\). \item[P2:] Minimize the one-dimensional Hausdorff measure over all compact connected sets \(A\) of the \(n\)-dimensional Euclidean space with the prescribed bound that \(F_M(A)\) is less or equal to a fixed positive real number \(r\). \end{itemize}} They prove that both problems P1 and P2 admit solutions and, in the case \(n=2\), that they have the same set of minimizers. Finally they prove that minimizers cannot contain closed loops and that they have some mild regularity properties.