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Given a metric space $(X,d)$ and a subset $S\subset X$, the projection $P_S$ onto $S$ is well-defined as a set valued function. I am interested in learning more about properties of these projections such as uniqueness, continuity, proximinality, etc.

Unfortunately, almost every paper I can find on this topic considers the special case where $X$ is a Banach space. I am interested specifically in the case where $X$ is a general metric space. Any pointers to papers or monographs would be appreciated.

Note. Apparently Ivan Singer has some papers from the 1960s on this, but I can’t track down the full text of the papers online. For example:

[1] Some remarks on approximative compactness. Rev. Roum. Math. Pures Appl. 9(1964), 167-177

References on metric spaces. Here are a couple papers I have found specifically regarding metric spaces:

[2] Khalil, R. (1988). Best approximation in metric spaces. Proceedings of the American mathematical society, 103(2), 579-586.

[3] Gupta, S., & Narang, T. D. (2017). STRONG PROXIMINALITY IN METRIC SPACES. Novi Sad J. Math, 47(2), 107-116.

References on Banach spaces. Just in case, here are useful some references on the Banach space version.

[4] D. D. Repovš and P.V. Semenov, Continuous Selections of Multivalued Mappings, Kluwer Academic Publishers, Dordrecht 1998. http://www.pef.uni-lj.si/repovs/knjige/kazalo_book.htm See Chapter C.6 on metric projections

[5] Alber, Y. I. (1996). A bound for the modulus of continuity for metric projections in a uniformly convex and uniformly smooth Banach space. journal of approximation theory, 85(3), 237-249.

[6] Deutsch, F., & Lambert, J. M. (1980). On continuity of metric projections. Journal of Approximation Theory, 29(2), 116-131.

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  • $\begingroup$ I'm not sure if this is what you want, but closest-point projection is coarsely well-defined (ie the image sets are of bounded diameter) when the metric space is "Gromov-hyperbolic". This is used a great deal in the geometric-group-theory literature. $\endgroup$ – HJRW Feb 27 at 17:10

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