Chebyshev centers and Chebyshev radius

Prove that for any bounded set in a metric space the inequalities hold

Prove that in the space $l_{2}(2)$ for every bounded set $M$ the Chebyshev center exists and $\operatorname {diam}(M) / 2 \leqslant r_{C}(M) \leqslant \operatorname {diam}(M) / \sqrt{3}$.

Prove that in the space $l_{\infty }(2)$ for every bounded set $M$ the Chebyshev center exists and $r_{C}(M) = \operatorname {diam}(M) / 2$.

Give an example of a metric space in which for every bounded set $M$ the equality $r_{C}(M) = \operatorname {diam}(M)$ holds.

Let $X$ be a metric compact. Prove that every subset of $X$ has a Chebyshev center.

(V. D. Erokhin). Prove that every compact of diameter $d$ in the space $C[0,1]$ lies in some closed ball of radius $d / 2$ (i.e., its Chebyshev radius equals $d / 2$). Prove that for Hilbert spaces (of dimension greater than 1) this is false.