Not too many miles from ${\mathrm{\ell }}_{\mathrm{\infty }}$ to $L_{\mathrm{\infty }}$

For either real or complex scalars, we improve the known upper bound for the Banach–Mazur distance between $L_{\mathrm{\infty }}[0,1]$ and ${\mathrm{\ell }}_{\mathrm{\infty }}$, proving $d_{\mathrm{BM}}(L_{\mathrm{\infty }}[0,1],{\mathrm{\ell }}_{\mathrm{\infty }})⩽(2+\sqrt{5}{)}^2=9+4\sqrt{5}=17.944\dots$. The additional input is a stabilised kernel estimate: the copy of ${\mathrm{\ell }}_{\mathrm{\infty }}$ in $L_{\mathrm{\infty }}$ is chosen so that its kernel is an ${\mathrm{\ell }}_{\mathrm{\infty }}$-sum of singular two-point hyperplanes, and Cohen’s bound-two construction is then applied coordinatewise. The quantitative part of the proof is an asymmetric Pełczyński-type factorisation, with the elementary optimisation included below.