Isometric approximation

Suppose that A subset of R-n is a bounded set of diameter 1 and that f: A - -> l(2) is a map satisfying the nearisometry condition \x - y\ - epsilon : less than or equal to \fx - fy\ less than or equal to \x - y\ + epsilon wit h epsilon less than or equal to 1. Then there is an isometry S: A --> l(2) such that \Sx - fx\ less than or equal to c(n) root epsilon for all x in A. If A satisfies a thickness condition and if f: A --> R-n, then there is an isometry S: R-n --> R-n with \Sx - fx\ less than or equal to c(n) epsilon /q, where q is a thickness parameter.