**Question:** Is there an example of a nonconvex Chebyshev set $S$ in a metric space $(X,d)$ whose projection map is continuous?


For convexity to be well-defined, we need to assume that $X$ is a vector space, but not necessarily a normed space. The projection map is defined as
$$
P\_Sx
= \text{argmin}\_{y\in S} d(x,y).
$$


A Chebyshev set is a set $S$ such that $P\_Sx$ is unique and hence well-defined for all $x\in S$.


It is a [classical result](https://www.ams.org/journals/tran/1968-134-02/S0002-9947-1968-0232146-1/S0002-9947-1968-0232146-1.pdf) that if $X$ is a Banach space and $d$ is the Banach space norm, then continuity of $P\_S$ is equivalent to convexity of $S$. Thus no example exists in a Banach space.


[1](https://www.ams.org/journals/tran/1968-134-02/S0002-9947-1968-0232146-1/S0002-9947-1968-0232146-1.pdf) Wulbert, D. E. (1968). Continuity of metric projections. Transactions of the American Mathematical Society, 134(2), 335-341.



Example of a nonconvex Chebyshev set in a metric space with continuous projection?
Let $E$ be the (incomplete) subspace of sequences in $\ell^{2}(\mathbb{R})$ having at most finitely many nonzeros terms. In [1], a subset $S$ of $E$ is constructed (by a long induction argument), which has the following properties :


* $S$ is closed and nonconvex,
* each point in $E$ has a unique nearest point in $S$,
* the projection is continuous.


This construction is mentioned in the book by D. Braess, Non linear approximation theory (p.42-43), Springer 1986.


[1] G. Johnson,
A nonconvex set which has the unique nearest point property, 
J. Approx. Theory 51 (1987), no. 4, 289-332. 


