An Overview of Maximal Distance Minimizers Problem

Consider a compact M subset Rd and l > 0. A maximal distance minimizer problem is to find a connected compact set Σ of the length (one-dimensional Hausdorff measure H1) at most l that minimizes maxy in M dist(y,Σ), where dist stands for the Euclidean distance. We give a survey on the results on the maximal distance minimizers and related problems. Also we fill some natural gaps by showing NP-hardness of the maximal distance minimizing problem, establishing its Γ-convergence, considering the penalized form and discussing uniqueness of a solution. We finish with open questions.