Proper Affine Isometric Actions On Banach Spaces

In this paper it is proved,that every locally compact,second countable group G has a left invariant metric d,which generates the topology on G,and which is proper,we denote such a metric a plig metric.Later we give the definition of d-balls have polynomial growth in the sense that μ（B_d（e,n））≤p（n）,（?）_n （?） N,and prove that every locally compact,second countable group has a plig metric,and a plig metric d can be chosen,such that the d-balls have polynomial growth.Moreover we obtain the following result:every locally compact,second countable group G admits a proper affine isometric action on the reflexive separable strictly convex Banach space X=（?）L²ⁿ（G,μ）（the direct sum is taken in the l~2-sense）,and the affine action can decay at any k speed（k≥2）,ie.for every g in G,‖bⁿ（g）_2n~2=o（1/n^k）is established.