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