By Theorem 1.6 in the book ["Geometric Nonlinear Functional Analysis" by Benyamini and Lindenstrauss](https://books.google.com.ua/books/about/Geometric_Nonlinear_Functional_Analysis.html?id=lXZ95EKwjYUC&redir_esc=y), 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



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> **Problem.** Give some upper bounds on the smallest Lipschitz constant $L$ of a retraction $r:\ell\_\infty[0,1]\to C[0,1]$. Is $L\le 20$?
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What is the smallest Lipschitz constant of a Lipschitz retraction of $\ell_\infty([0,1])$ onto $C[0,1]$?
Yes, as we have this theorem of Nigel Kalton:



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


