Geodesic
Metric projection onto subsets of compact connected two-dimensional Riemannian manifolds
Vestnik Moskovskogo universiteta. Matematika, mehanika, no. 4 (2017), pp. 15-20 
K. S. Shklyaev. Metric projection onto subsets of compact connected two-dimensional Riemannian manifolds. Vestnik Moskovskogo universiteta. Matematika, mehanika, no. 4 (2017), pp. 15-20. http://geodesic.mathdoc.fr/item/VMUMM_2017_4_a1/
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     author = {K. S. Shklyaev},
     title = {Metric projection onto subsets of compact connected two-dimensional {Riemannian} manifolds},
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     pages = {15--20},
     year = {2017},
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     url = {http://geodesic.mathdoc.fr/item/VMUMM_2017_4_a1/}
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Voir la notice de l'article provenant de la source Math-Net.Ru

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.

[1] Karlov M.I., “Chebyshevskie mnozhestva na mnogoobraziyakh”, Tr. IMM UrO RAN, 4, 1996, 157–161 | Zbl

[2] Gromol D., Klingenberg V., Meier V., Rimanova geometriya v tselom, Mir, M., 1971

[3] Mischenko A.S., Fomenko A.T., Kurs differentsialnoi geometrii i topologii, Faktorial Press, M., 2000 | MR

[4] Skopenkov A.B., Algebraicheskaya topologiya s geometricheskoi tochki zreniya, MTsNMO, M., 2015