ON LIPSCHITZ EXTENSION FROM FINITE SUBSETS

Abstract

We prove that for every n is an element of N there exists a metric space (X, d(X)), an n-point subset S subset of X, a Banach space (Z, parallel to . parallel to(Z)) and a 1-Lipschitz function f : S -> Z such that the Lipschitz constant of every function F : X -> Z that extends f is at least a constant multiple of root log n. This improves a bound of Johnson and Lindenstrauss [JL84]. We also obtain the following quantitative counterpart to a classical extension theorem of Minty [Min70]. For every alpha is an element of(1/2, 1] and n is an element of N there exists a metric space (X, d(X)), an n-point subset S subset of X and a function f : S -> l(2) that is alpha-Holder with constant 1, yet the alpha-Holder constant of any F : X -> l(2) that extends f satisfies parallel to F parallel to(Lip(alpha)) greater than or similar to (log n) (2 alpha - 1/4 alpha) + (log n/log log n) (alpha 2 -) (1/2). We formulate a conjecture whose positive solution would strengthen Ball’s nonlinear Maurey extension theorem [Bal92], serving as a far-reaching nonlinear version of a theorem of K “ onig, Retherford and Tomczak-Jaegermann [KRTJ80]. We explain how this conjecture would imply as special cases answers to longstanding open questions of Johnson and Lindenstrauss [JL84] and Kalton [Kal04].