Around Ulam's question on retractions (Q1947838)

Ulam's question on retractions, raised by S.\, Ulam around 1935, is: ``Can one transform continously the solid sphere of a Hilbert space into its boundary such that the transformation should be the identity on the boundary of the ball?'' An equivalent formulation of the problem is: ``If \(H\) is a Hilbert space, \(B\) its unit ball and \(S\) the unit sphere, is \(S\) the retract of \(B\)?'' Due to \textit{S. Kakutani} [Proc. Imp. Acad. Tokyo 19, 269--271 (1943; Zbl 0060.27701)], the answer is yes, via a Lipschitzian mapping. The problem is now how small is the Lipschitz constant? This is known as the optimal retraction problem. In this paper, the authors propose a new approach in the study of the above problem.