On the extension of Lipschitz, Lipschitz-Hölder continuous, and monotone functions
George J. Minty · Bulletin of the American Mathematical Society · 1970
Introduction. The well-known theorem of Kirszbraun [9], [14]asserts that a Lipschitz function from R n to itself, with domain a finite point-set, can be extended to a larger domain including any arbitrarily chosen point.(The Euclidean norm is essential; see Schönbeck [lö], Grünbaum [8].)This theorem was rediscovered by Valentine [17] using different methods.The writer [12] proved the same fact for a "monotone" function, and Grünbaum [9] combined these two theorems into one.A further improvement to the writer's theorem was given by Debrunner and Flor [ó], who showed that the desired new functional value could always be chosen in the convex hull of the given functional values; several different proofs of this fact have now been given (see [14], [3]).An easy consequence of Kirszbraun's theorem is that a Lipschitz function in Hubert space with maximal domain is everywhere-defined (see [ll], [13]).It was shown by S. Banach [l] that a real-valued function defined on a subset of a metric space and satisfying \f(yi)-f(y2)\ = [5(^i, 3>2)] a , with 0<a^l (we call this "Lipschitz-Hölder continuity"), can be extended to the whole metric space so as to satisfy the same inequality.Banach's theorem was rediscovered by Czipszer and Gehér [4] in case ce = l (but note that Banach's result follows, since [8(yi, 3/2)]" is another metric if ce^l).For a general review of the above subjects, see the article of Danzer, Grünbaum, and Klee [5]; see also [7].In this paper, we give a unified method for proving all the above results, and also new theorems, the most striking of which is the following generalization of the Kirszbraun and Banach theorems: THEOREM 1.Let H be a Hilbert space, M a metric space, DC.M. Suppose f :D-*H satisfies \\f(yi)-f(y2)\\ S [ô(y u y 2 ) ] a (0<a^l).Then there exists an extension off to all of M satisfying the same inequality, if either (i) a^h or A MS Subject Classifications.