On Lipschitz extension from finite subsets

We prove that for every n ∈ ℕ there exists a metric space (X, d_X), an n-point subset S ⊆ X, a Banach space (Z, ‖ · ‖ _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 logn. 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 α ∈ (1/2, 1] and n ∈ ℕ there exists a metric space (X, d_X), an n-point subset S ⊆ X and a function f: S → ℓ₂ that is α-Hölder with constant 1, yet the α-Hölder constant of any F: X → ℓ₂ that extends f satisfies ‖F‖Lip(α)>(logn)2α−14α+(lognloglogn)α2−12. 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 König, 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].