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A 5/8 Lower Bound on the Banach--Mazur Distance to the Cross-Polytope
Abstract
Let δ>0 be fixed, let m = ⌈(1+δ)n⌉, and let Γ be an n× m matrix with independent standard Gaussian entries. For the Gaussian Gluskin polytope Gm = Γ(B1m) we prove p{dBM(Gm,B1n) ≥ cδn5/8(log n)−1/8} ≥ 1−Ce−cn. Consequently, for all sufficiently large n, R1(n)≥ c n5/8(log n)−1/8. The proof uses an exact coefficient quantization, a simultaneous Gaussian compression in coefficient quotients, and an inverse-trace estimate for arbitrary inscribed bases.
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BibTeX
@misc{friedland2026lower,
title = {{A $5/8$ Lower Bound on the Banach--Mazur Distance to the Cross-Polytope}},
author = {Omer Friedland},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2608.17743},
eprint = {2608.17743},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.17743},
}
RIS
TY - UNPB
TI - A 5/8 Lower Bound on the Banach--Mazur Distance to the Cross-Polytope
AU - Omer Friedland
PY - 2026
DA - 2026/09/27
PB - arXiv
DO - 10.48550/arxiv.2608.17743
UR - https://arxiv.org/abs/2608.17743
AB - Let δ>0 be fixed, let m = ⌈(1+δ)n⌉, and let Γ be an n× m matrix with independent standard Gaussian entries. For the Gaussian Gluskin polytope G_m = Γ(B_1^m) we prove p{d_BM(G_m,B_1^n) ≥ c_δn^5/8(log n)^−1/8} ≥ 1−Ce^−cn. Consequently, for all sufficiently large n, R_1(n)≥ c n^5/8(log n)^−1/8. The proof uses an exact coefficient quantization, a simultaneous Gaussian compression in coefficient quotients, and an inverse-trace estimate for arbitrary inscribed bases.
ER -