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) .
Abstract. Let H be an at least two-dimensional real Hilbert space with the unit sphere S H . For α ∈ [−1, 1] and n ∈ S H we define an (α, n)-spherical cap by Sα,n = {x ∈ S H : x, n ≥ α}. We show that the distance between the set of contractions T : Sα,n → Sα,n and the identity mapping is positive iff α < 0. We also study the fixed point property and the minimal displacement problem in this setting for nonexpansive mappings.
It is shown that in many instances the fixed point property for nonexpansive mappings actually implies the fixed point property for a strictly larger family of mappings. This paper is largely expository, but some of the observations are not readily available, and some appear here for the first time. Several related open questions in are discussed. The emphasis is on accessible problems, especially those that require little background. The problems themselves have been given little thought and may be trivial or difficult.
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