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The sharp exponent for the Banach--Mazur distance to the cube
Abstract
Let R∞(n) be the maximal Banach--Mazur distance from an n-dimensional real normed space to ℓ∞n. We prove that R∞(n)≥ cn2/3(log(2n))−10. Together with the upper bound of Friedland, Naor, and Youssef, this gives R∞(n) = n2/3+o(1). The examples are polars of symmetric Gaussian polytopes with 2n generators, and the lower bound holds with exponentially high probability. A joint minimization over coefficient representations and positive weights yields an exact second-variation identity and sparse coefficients. A Gaussian projection argument controls all low-rank completions of the recorded matrix. An adaptive choice of exceptional generators then reduces the remaining case to a fixed-shift Gaussian small-ball estimate over two finite nets.
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BibTeX
@misc{friedland2026sharp,
title = {{The sharp exponent for the Banach--Mazur distance to the cube}},
author = {Omer Friedland},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.36917},
eprint = {2610.36917},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.36917},
}
RIS
TY - UNPB
TI - The sharp exponent for the Banach--Mazur distance to the cube
AU - Omer Friedland
PY - 2026
DA - 2026/10/27
PB - arXiv
DO - 10.48550/arxiv.2610.36917
UR - https://arxiv.org/abs/2610.36917
AB - Let R_∞(n) be the maximal Banach--Mazur distance from an n-dimensional real normed space to ℓ_∞^n. We prove that R_∞(n)≥ cn^2/3(log(2n))^−10. Together with the upper bound of Friedland, Naor, and Youssef, this gives R_∞(n) = n^2/3+o(1). The examples are polars of symmetric Gaussian polytopes with 2n generators, and the lower bound holds with exponentially high probability. A joint minimization over coefficient representations and positive weights yields an exact second-variation identity and sparse coefficients. A Gaussian projection argument controls all low-rank completions of the recorded matrix. An adaptive choice of exceptional generators then reduces the remaining case to a fixed-shift Gaussian small-ball estimate over two finite nets.
ER -