We prove that, in every infinite dimensional Hilbert space, there exists $t_0>-1$ such that the smallest Lipscthiz constant of retractions from the unit ball onto its boundary is the same as the smallest Lipschitz constant of retractions from the unit ball onto its $t$-spherical cup for all $t\in[-1,t_0]$.
"Lipschitz retractions onto sphere vs spherical cup in a Hilbert space." Topol. Methods Nonlinear Anal. 52 (2) 677 - 691, 2018. https://doi.org/10.12775/TMNA.2018.034