  We show that if $X$ is a complete metric space with uniform relative normal
structure and $G$ is a subgroup of the isometry group of $X$ with bounded
orbits, then there is a point in $X$ fixed by every isometry in $G$. As a
corollary, we obtain a theorem of U. Lang (2013) concerning injective metric
spaces. A few applications of this theorem are given to the problems of inner
derivations. In particular, we show that if $L_{1}(\mu )$ is an essential
Banach $L_{1}(G)$-bimodule, then any continuous derivation $\delta
:L_{1}(G)\rightarrow L_{\infty }(\mu )$ is inner. This extends a theorem of B.
E. Johnson (1991) asserting that the convolution algebra $L_{1}(G)$ is weakly
amenable if $G$ is a locally compact group.
