Growth rate of Lipschitz constants for retractions between finite subset spaces

For any metric space X, finite subset spaces of X provide a sequence of isometric embeddings X = X(1) ⊂ X(2) ⊂ · · ·. The existence of Lipschitz retractions rn : X(n) → X(n-1) depends on the geometry of X in a subtle way. Such retractions are known to exist when X is an Hadamard space or a finite-dimensional normed space. But even in these cases it was unknown whether the sequence {rn} can be uniformly Lipschitz. We give a negative answer by proving that Lip(rn) must grow with n when X is a normed space or an Hadamard space.