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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 $L\le 20$?

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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.

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  • $\begingroup$ @TarasBanakh, you are most welcome. I remember that I had used this result some time ago. $\endgroup$
    – Tomasz Kania
    Commented Apr 22, 2018 at 16:01
  • $\begingroup$ On the first page of this paper the constant 20 (by Lindenstrauss) is mentioned. It is the same as was asked by OP. May I wonder, is it something special to have Lipschitz constant 20? $\endgroup$
    – Fedor Petrov
    Commented Apr 22, 2018 at 16:04
  • $\begingroup$ @FedorPetrov, I think it simply the estimate that follows from the Lindenstrauss' (non-optimal) proof. Kalton wanted to make a case that his result indeed strengthens the original result. $\endgroup$
    – Tomasz Kania
    Commented Apr 22, 2018 at 16:05
  • $\begingroup$ @FedorPetrov The constant 20 appears in the book of Benyamini and Lindenstrauss, where they construct a uniformly continuous retraction of $\ell_\infty(K)$ onto $C(K)$ which is locally Lipschitz with constant 20 at a neighborhood of $C(K)$ in $\ell_\infty(K)$. But even this local constant 20 cannot be traced to give a reasonable global Lipschitz constant. So the result of Kalton is very nice and exact (2 cannot be further improved). $\endgroup$
    – Taras Banakh
    Commented Apr 22, 2018 at 16:22

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