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. 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) .

On Minimal Displacement Problem and Retractions of Balls Onto Spheres

“…Similar examples can be constructed in many spaces as, for example, c 0 , l ∞ , spaces of differentiable functions with standard norms, and in all subspaces of C[a, b] of finite codimension. This last recent result is not published yet and is due to Bolibok [3].…”

“…The estimate (3) shows that d(L) is not achieved on any T ∈ L. This is a partial observation of the fact that for any continuous mapping T : B → B we have d(T ) < 1 [3].…”

Optimal retraction problem for proper $k$-ball-contractive mappings in $C^m [0,1]$

“…There is no space X for which the exact value of k 0 (X) is known, for a survey on the subject we refer to [13,18,19] and bibliography therein. The universal known bound from below is k 0 (X) ≥ 3; for some spaces there are better estimates, for example, k 0 (H) ≥ 4.58 for Hilbert space H (see [14]), k 0 (l 1 ) ≥ 4 (see [6]).…”