Metric projection onto subsets of compact connected two-dimensional Riemannian manifolds

The paper is focused on combinatorial properties of the metric projection $P_{E}$ of a compact connected Riemannian two-dimensional manifold $M^{2}$ onto its subset $E$ consisting of $k$ closed connected sets $E_{j}$. The point $x \in M^{2}$ is called exceptional if $P_{E}(x)$ contains points from no less than three different $E_{j}$. The sharp estimate for the number of exceptional points is obtained in terms of $k$ and the type of the manifold $M^{2}$. Similar estimate is proved for finitely connected subsets $E$ of a normed plane.