Tight Stability Estimates Near the Simplex and Applications for the Banach-Mazur Distance and Rogers-Shephard-Type Inequalities
Abstract
We establish a tight stability estimate for the Minkowski asymmetry near its maximal value, improving earlier results in both the range for the admissible error and the strength of the estimate. More precisely, if an n-dimensional convex body K has Minkowski asymmetry s(K) ≥ n−ε for ε ∈ [0,1), then its Banach-Mazur distance to the n-simplex is at most 1 + ε + ε2/(2(1−ε)). This dimension-independent estimate is sharp to the linear order in ε, including the constant. We apply this estimate to several problems. First, we prove a stability result for the maximal Banach-Mazur distance to the Euclidean ball, improving a previous estimate to the optimal linear order. As a key ingredient, we verify the conjecture that every convex body K contains a translated copy of its volume-minimal circumscribed ellipsoid scaled down by a factor √(n s(K)). Second, we prove a sharp common generalization of Schneider's higher-order Rogers-Shephard inequality and the Lp-Rogers-Shephard inequality, and establish a stability result of the optimal linear order. These results are based on the recent positive answer to the inequality part of the higher-order Godbersen conjecture and the accompanying proof of the Lp-Rogers-Shephard inequality. Finally, we improve upper bounds for the diameter of the Banach-Mazur compactum in fixed dimensions.
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BibTeX
@misc{brandenberg2026tight,
title = {{Tight Stability Estimates Near the Simplex and Applications for the Banach-Mazur Distance and Rogers-Shephard-Type Inequalities}},
author = {René Brandenberg and Bernardo González Merino and Florian Grundbacher},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2607.27041},
eprint = {2607.27041},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.27041},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Tight Stability Estimates Near the Simplex and Applications for the Banach-Mazur Distance and Rogers-Shephard-Type Inequalities
AU - René Brandenberg
AU - Bernardo González Merino
AU - Florian Grundbacher
PY - 2026
DA - 2026/09/14
PB - arXiv
DO - 10.48550/arxiv.2607.27041
UR - https://arxiv.org/abs/2607.27041
AB - We establish a tight stability estimate for the Minkowski asymmetry near its maximal value, improving earlier results in both the range for the admissible error and the strength of the estimate. More precisely, if an n-dimensional convex body K has Minkowski asymmetry s(K) ≥ n−ε for ε ∈ [0,1), then its Banach-Mazur distance to the n-simplex is at most 1 + ε + ε^2/(2(1−ε)). This dimension-independent estimate is sharp to the linear order in ε, including the constant. We apply this estimate to several problems. First, we prove a stability result for the maximal Banach-Mazur distance to the Euclidean ball, improving a previous estimate to the optimal linear order. As a key ingredient, we verify the conjecture that every convex body K contains a translated copy of its volume-minimal circumscribed ellipsoid scaled down by a factor √(n s(K)). Second, we prove a sharp common generalization of Schneider's higher-order Rogers-Shephard inequality and the L_p-Rogers-Shephard inequality, and establish a stability result of the optimal linear order. These results are based on the recent positive answer to the inequality part of the higher-order Godbersen conjecture and the accompanying proof of the L_p-Rogers-Shephard inequality. Finally, we improve upper bounds for the diameter of the Banach-Mazur compactum in fixed dimensions.
ER -