Gaussian polytopes with large Banach-Mazur distance to the cross-polytope
Abstract
Let B1n be the standard cross-polytope in ℝn, let g1,…,gm be independent standard Gaussian vectors in ℝn, and set Gm=conv± g1,…,± gm. For m=n3 it is proved that ℙ{dBM(Gm,B1n)geqslant c n5/8(ln n)−5/8}geqslant 1−2/n for a suitable absolute constant c>0. This independently improves the polynomial exponent 4/7 in Friedland's preceding work. Independent concurrent work of Friedland, which appeared after completion of the present manuscript, obtains the same polynomial exponent with the stronger logarithmic factor (ln n)−1/4 by a different argument. The proof uses Friedland's discretization and conditioning argument together with the K/U decomposition. A selected family of K vectors is suppressed and the remaining K vectors are quotiented out. In the resulting quotient simultaneous bounds are proved for every top-dimensional exterior product formed from the suppressed K vectors and the U vectors. A Dvoretzky-Rogers selection after L"owner normalization converts these determinant estimates into a bound for the minimum volume ellipsoid of the whole projected polytope and Maurey's empirical method then gives the required Gaussian measure estimate.
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BibTeX
@misc{hmadi2026gaussian,
title = {{Gaussian polytopes with large Banach-Mazur distance to the cross-polytope}},
author = {Antonios Hmadi},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.27796},
eprint = {2609.27796},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.27796},
note = {CC BY-NC-SA 4.0},
}
RIS
TY - UNPB
TI - Gaussian polytopes with large Banach-Mazur distance to the cross-polytope
AU - Antonios Hmadi
PY - 2026
DA - 2026/09/24
PB - arXiv
DO - 10.48550/arxiv.2609.27796
UR - https://arxiv.org/abs/2609.27796
AB - Let B_1^n be the standard cross-polytope in ℝ^n, let g_1,…,g_m be independent standard Gaussian vectors in ℝ^n, and set G_m=conv± g_1,…,± g_m. For m=n^3 it is proved that ℙ{d_BM(G_m,B_1^n)geqslant c n^5/8(ln n)^−5/8}geqslant 1−2/n for a suitable absolute constant c>0. This independently improves the polynomial exponent 4/7 in Friedland's preceding work. Independent concurrent work of Friedland, which appeared after completion of the present manuscript, obtains the same polynomial exponent with the stronger logarithmic factor (ln n)^−1/4 by a different argument. The proof uses Friedland's discretization and conditioning argument together with the K/U decomposition. A selected family of K vectors is suppressed and the remaining K vectors are quotiented out. In the resulting quotient simultaneous bounds are proved for every top-dimensional exterior product formed from the suppressed K vectors and the U vectors. A Dvoretzky-Rogers selection after L"owner normalization converts these determinant estimates into a bound for the minimum volume ellipsoid of the whole projected polytope and Maurey's empirical method then gives the required Gaussian measure estimate.
ER -