Lipschitzian retracts and regular retractions on sets in Banach spaces

"A subset K of a Banach space X is a lipschitzian retract of X if there exists a mapping R : X ! K satisfying the Lipschitz condition kRx ¡ Ryk · kkx ¡ yk with certain constant k and such that Rx = x for all x 2 K: It is a known fact that the unit ball B ½ X is the lipschitzian retract of X with Lipschitz constant k = 2: The standard retraction is the radial projection P defined by Px = ½ x if kxk · 1 x kxk if kxk > 1 : However in some spaces there are other retractions onto B with smaller Lipschitz constant. For example in any Hilbert space H, not only the unit ball but also any nonempty closed and convex subset C ½ H is a nonexpansive (meaning lipschitzian with k = 1 ) retract of H: The less known fact is that in infinitely dimensional Banach spaces, the unit sphere S is a lipschitzian retract of X: However, the Lipschitz constant of any retraction R : X ! S has to be sufficiently large. The standard evaluation shows k ¸ 3: For some regular spaces it has to be even higher. The problem of finding lipschitzian retractions of X onto S having smallest possible constant k is known to specialists as optimal retraction problem. There is no space X for which it is solved yet. In spite of slow, steady progress in recent years, there are only some rough evaluations. For example there are spaces (l1