Year: 2026
Type: preprint
Abstract: For either real or complex scalars, we improve the known upper bound for the Banach--Mazur distance between $L_\infty[0,1]$ and $\ell_\infty$, proving $d_{\operatorname{BM}}(L_\infty[0,1],\ell_\infty)\leqslant (2+\sqrt5)^2=9+4\sqrt5=17.944\ldots$. The additional input is a stabilised kernel estimate: the copy of $\ell_\infty$ in $L_\infty$ is chosen so that its kernel is an $\ell_\infty$-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.
Source: arXiv (Cornell University)
Author: Kania, Tomasz
Language: English
Topic: Advanced Banach Space Theory
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7221790105