PreprintCC BY 4.0arXiv
When the noncommutative AM-GM inequality holds
Abstract
In this note, we prove that the noncommutative AM-GM inequality holds if n≥ 2⌈ m/2 ⌉2. The motivation comes from counterexamples constructed in [De Sa, Random reshuffling is not always better, NeurIPS2020]. The proof constructs a vertex measure based on the Chebyshev nodes on the Boolean cube to extract the distinct indices. The main difficulty is that the measure is not positive on non-integer nodes. The key technique comes from [Grigoriev, Complexity of Positivstellensatz proofs for the knapsack, Computational Complexity (2001)] and eventually transforms the problem into a quadrature estimate.
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BibTeX
@misc{zhong2026when,
title = {{When the noncommutative AM-GM inequality holds}},
author = {Yimin Zhong},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.04874},
eprint = {2610.04874},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.04874},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - When the noncommutative AM-GM inequality holds
AU - Yimin Zhong
PY - 2026
DA - 2026/10/04
PB - arXiv
DO - 10.48550/arxiv.2610.04874
UR - https://arxiv.org/abs/2610.04874
AB - In this note, we prove that the noncommutative AM-GM inequality holds if n≥ 2⌈ m/2 ⌉^2. The motivation comes from counterexamples constructed in [De Sa, Random reshuffling is not always better, NeurIPS2020]. The proof constructs a vertex measure based on the Chebyshev nodes on the Boolean cube to extract the distinct indices. The main difficulty is that the measure is not positive on non-integer nodes. The key technique comes from [Grigoriev, Complexity of Positivstellensatz proofs for the knapsack, Computational Complexity (2001)] and eventually transforms the problem into a quadrature estimate.
ER -