Geodesic
Sets with not more than two-valued metric projection on planes
Vestnik Moskovskogo universiteta. Matematika, mehanika, no. 6 (2013), pp. 14-19

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For a set $M$ in the Euclidean plane $\mathbb{R}^2$, we prove that if any point $x\in\mathbb{R}^2$ has one or two closest points in $M$, then each point of the convex hull of $M$ lies in the segment with endpoints in $M$. 
@article{VMUMM_2013_6_a2,
     author = {A. A. Flerov},
     title = {Sets with not more than two-valued metric projection on planes},
     journal = {Vestnik Moskovskogo universiteta. Matematika, mehanika},
     pages = {14--19},
     publisher = {mathdoc},
     number = {6},
     year = {2013},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/VMUMM_2013_6_a2/}
}
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A. A. Flerov. Sets with not more than two-valued metric projection on planes. Vestnik Moskovskogo universiteta. Matematika, mehanika, no. 6 (2013), pp. 14-19. http://geodesic.mathdoc.fr/item/VMUMM_2013_6_a2/