Isometric approximation

Suppose thatA ⊂Rn is a bounded set of diameter 1 and that:f:A →l2 is a map satisfying the nearisometry condition |x−y|−ɛ≤|fx−fy|≤|x−y|+ɛ withɛ≤1. Then there is an isometryS:A →l2 such that |Sx−fx|≤cn √ɛ for allx inA. IfA satisfies a thickness condition and iff:A →Rn, then there is an isometryS:Rn →Rn with |Sx−fx|≤cnɛ/q, whereq is a thickness parameter.