TITLE: What is the smallest Lipschitz constant of a Lipschitz retraction of $\ell_\infty([0,1])$ onto $C[0,1]$?
QUESTION [7 upvotes]: By Theorem 1.6 in the book "Geometric Nonlinear Functional Analysis" by Benyamini and Lindenstrauss, the Banach space $C[0,1]$ is a Lipschitz retract of the Banach space $\ell_\infty[0,1]$. Unfortunately, the proof does not give any upper bounds on possible Lipschitz constants of the retraction. So, the

Problem. Give some upper bounds on the smallest Lipschitz constant $L$ of a retraction $r:\ell_\infty[0,1]\to C[0,1]$. Is $L\le 20$?

REPLY [11 votes]: Yes, as we have this theorem of Nigel Kalton:

Let $K$ be a compact metric space. Then $C(K)$ is an absolute 2-Lipschitz retract.

Please see [1] for details.
[1] Kalton, Nigel J. "Extending Lipschitz maps into C (K)-spaces." Israel Journal of Mathematics 162.1 (2007): 275-315.