Injectivity of quasi-isometric mappings of balls

Abstract

Let X and Y be real Banach spaces. A mapping f of an open subset R of X into Y is said to be (m, M)-isometric if it is a local homeomorphism for which M > D+ f(x) and m D+ f (x) and 0 < m < D-f (x) for all x in R, where D+f(x) and D-f(x) are, respectively, the upper and lower limits of If(y) f(x)I/Iy xI as y -+ x. Less precisely, f is called quasi-isometric if it is (m, M)-isometric for some m, M. We cite two results of John as THEOREM 1. Let X be a Hilbert space and let Y be a Banach space. Let R C X be an open ball and let f: R -+ Y be an (m, M)-isometric mapping. Then either one of the following conditions implies that f is one-to-one: (A) M/m < ((1 + V5-)/2)1/2 1.272... (see [1]), (B) Y is also a Hilbert space and M/m < X = 1.414... (see [3]). John's proofs of these injectivity criteria use properties of Hilbert spaces which have no analogue in general Banach spaces. The purpose of this paper is to show that even so it is possible to derive similar criteria valid in the more general context. We shall also show that the constant ((1 + V5-)/2)1/2 of (A) may be replaced by X and that the constant V of (B) may be replaced by (1 + X/-)1/2 1.553.... Let ,O = 1.114... denote the (unique) real root of the equation (1) = x + (25X2 -8x) / (1) X ~~~~2(3X2 -X) We shall prove THEOREM 2. Let X and Y be Banach spaces. Let R C X be an open ball and let f: R -+ Y be an (m, M)-isometric mapping. Then any one of the following conditions implies that f is one-to-one: (C) M/m < to, (D) X is a Hilbert space and M/m < V2-, (E) both X and Y are Hilbert spaces and M/m < (1 + V)1/2. Received by the editors August 15, 1981. 1980 Mathematics Subject Classification. Primary 46B99, 47H99.