Minimal displacement and retraction problems in infinite-dimensional Hilbert spaces

Abstract

We give the first constructive example of a Lipschitz mapping with positive minimal displacement in an infinite-dimensional Hilbert space H . H. We use this construction to obtain an evaluation from below of the minimal displacement characteristic in the space H . H. In the second part we present a simple and constructive proof of existence of a Lipschitz retraction from a unit ball B B onto a unit sphere S S in the space H H , and we improve an evaluation from above of a retraction constant k 0 ( H ) . k_{0}\left ( H\right ) .