PreprintCC BY 4.0arXiv
Not too many miles from ℓ∞ to L∞
Abstract
For either real or complex scalars, we improve the known upper bound for the Banach--Mazur distance between L∞[0,1] and ℓ∞, proving dBM(L∞[0,1],ℓ∞)leqslant (2+√5)2=9+4√5=17.944…. The additional input is a stabilised kernel estimate: the copy of ℓ∞ in L∞ is chosen so that its kernel is an ℓ∞-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.
Related
- A Compressible Miles Instability for Wind-Wave Generation in a Neutral Atmosphere
- Smoothly knotted surfaces that remain distinct after many internal stabilizations
- Every Nonrecursive Many-One Degree Contains Either One or Infinitely Many Finite-One Degrees
- Regression adjustment in completely randomized experiments with many covariates
- Improved bounds in Birch's theorem for forms in many variables
BibTeX
@misc{kania2026many,
title = {{Not too many miles from $\ell_\infty$ to $L_\infty$}},
author = {Tomasz Kania},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.14121},
eprint = {2610.14121},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.14121},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Not too many miles from ℓ_∞ to L_∞
AU - Tomasz Kania
PY - 2026
DA - 2026/10/09
PB - arXiv
DO - 10.48550/arxiv.2610.14121
UR - https://arxiv.org/abs/2610.14121
AB - For either real or complex scalars, we improve the known upper bound for the Banach--Mazur distance between L_∞[0,1] and ℓ_∞, proving d_BM(L_∞[0,1],ℓ_∞)leqslant (2+√5)^2=9+4√5=17.944…. The additional input is a stabilised kernel estimate: the copy of ℓ_∞ in L_∞ is chosen so that its kernel is an ℓ_∞-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.
ER -