In this article it is proved, that every locally compact second countable group has a left invariant metric d, which generates the topology on G, and which is proper, ie. every closed d-bounded set in G is compact. Moreover, we obtain the following extension of a result due to N. Brown and E. Guentner: Every locally compact second countable $G$ admits a proper affine action on the reflexive and strictly convex Banach space $\bigoplus^{ \infty}_{ n=1} L^{ 2n}(G, d\mu),$ where the direct sum is taken in the $l^2$-sense.
Categories: math.OA