Leave $ (X, d) $ be a cat$ (0) $ space, $ {x_n } subset X $ be limited and $ K subset X $ be closed and convex Define $ varphi: , X longrightarrow mathbb {R}, $ by $ varphi (x) = limsup limits_ {n to infty} d (x, x_n) $ for each $ x in X. $ Then, there is a single point. $ u in K $ such that $$ varphi (u) = inf limits_ {x in K} varphi (x). $$

**Test**

Leave $ r = inf limits_ {x in K} varphi (x) $ Y $ epsilon> 0. $ Then, there is $ x_0 in K $ such that $ varphi (x_0) <r + epsilon. $ This implies that there is $ N in mathbb {N} $ such that

$$ x_0 in bigcup_ {n geq N} left ( bigcap_ {k = n} B (x_k, r + epsilon) cap K right) subset bigcup_ {n geq 1} left ( bigcap_ {k = n} B (x_k, r + epsilon) cap K right). $$

Define $ c _ { epsilon}: = bigcup_ {n geq 1} left ( bigcap_ {k = n} B (x_k, r + epsilon) cap K right). $ It is clear that $ c _ { epsilon} $ It is convex

**Question:** How is it $ overline {c _ { epsilon}} $ convex and $ c: = bigcap _ { epsilon> 0} overline {c _ { epsilon}} neq emptyset? $

**Details:** The current article that I am reviewing is Dhompongsa et al. To show that $ overline {c _ { epsilon}} $ it is convex, Dhompongsa et al., referring to Proposition 1.4 (1). I just have a hard time understanding it. Any help, please?