# What is the smallest Lipschitz constant of a Lipschitz retraction of $\ell_\infty([0,1])$ onto $C[0,1]$?

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 $L\le 20$?

Let $K$ be a compact metric space. Then $C(K)$ is an absolute 2-Lipschitz retract.
• @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). – Taras Banakh Apr 22 '18 at 16:22