An overview of maximal distance minimizers problem

Consider a compact $M\subset {\mathbb{R}}^d$ and $l>0$. A maximal distance minimizer problem is to find a connected compact set $\mathrm{\Sigma }$ of the length (one-dimensional Hausdorff measure $\mathcal{H}$) at most $l$ that minimizes $$\underset{y\in M}{\max }dist(y,\mathrm{\Sigma }),$$ 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 $\mathrm{\Gamma }$-convergence, considering the penalized form and discussing uniqueness of a solution. We finish with open questions.