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Sharp Metric Cotype Inequalities for L1 via Nonlinear Cut Smoothing

  • Qingjin Cheng
  • Yue Wang
  • Bo Xiang
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Abstract

For every 1≤ p<∞ and even integer m≥4, we determine the optimal order of the p-moment torus inequality for L1: it is mp n(1−p/2)++n, with comparison constants independent of p,m,n after taking p-th roots. Consequently, for every 2≤ q<∞ and 1≤ p≤ q, the corresponding metric cotype inequality holds at the sharp scale m=O(n1/q). In particular, the quadratic inequality holds with m=O(√n), answering the L1 sharp metric cotype question of Mendel and Naor. The proof combines finite cut representations with a nonlinear smoothing estimate and an exact ternary rounding identity.

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Cite

BibTeXRIS
BibTeX
@misc{cheng2026sharp,
  title = {{Sharp Metric Cotype Inequalities for $L_1$ via Nonlinear Cut Smoothing}},
  author = {Qingjin Cheng and Yue Wang and Bo Xiang},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.08749},
  eprint = {2609.08749},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.08749},
}
RIS
TY  - UNPB
TI  - Sharp Metric Cotype Inequalities for L_1 via Nonlinear Cut Smoothing
AU  - Qingjin Cheng
AU  - Yue Wang
AU  - Bo Xiang
PY  - 2026
DA  - 2026/09/09
PB  - arXiv
DO  - 10.48550/arxiv.2609.08749
UR  - https://arxiv.org/abs/2609.08749
AB  - For every 1≤ p<∞ and even integer m≥4, we determine the optimal order of the p-moment torus inequality for L_1: it is m^p n^(1−p/2)_++n, with comparison constants independent of p,m,n after taking p-th roots. Consequently, for every 2≤ q<∞ and 1≤ p≤ q, the corresponding metric cotype inequality holds at the sharp scale m=O(n^1/q). In particular, the quadratic inequality holds with m=O(√n), answering the L_1 sharp metric cotype question of Mendel and Naor. The proof combines finite cut representations with a nonlinear smoothing estimate and an exact ternary rounding identity.
ER  -