PreprintCC BY-NC-ND 4.0arXiv
Very tight embeddings of subspaces of Lp for 0<p<1
Abstract
For 0<p<1 and μ>1, every d-dimensional subspace of Lp embeds linearly into ℓp⌊μd⌋ with distortion at most C(p,μ)(log(d+2))1/p. If its Rademacher type q constant is at most T, where p<q≤2, the distortion is at most C(p,q,μ)T, independently of d. We also give explicit bounds as μ↓1. This answers the question of Johnson and Schechtman concerning the extension of their very tight embedding theorem below exponent one. The proof combines compression at a lower exponent with a change of density that preserves the number of coordinates, and applies over both scalar fields.
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BibTeX
@misc{cai2026very,
title = {{Very tight embeddings of subspaces of \(L_p\) for \(0<p<1\)}},
author = {Gang Cai and Longfa Sun},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.30546},
eprint = {2610.30546},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.30546},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Very tight embeddings of subspaces of L_p for 0<p<1
AU - Gang Cai
AU - Longfa Sun
PY - 2026
DA - 2026/10/22
PB - arXiv
DO - 10.48550/arxiv.2610.30546
UR - https://arxiv.org/abs/2610.30546
AB - For 0<p<1 and μ>1, every d-dimensional subspace of L_p embeds linearly into ℓ_p^⌊μd⌋ with distortion at most C(p,μ)(log(d+2))^1/p. If its Rademacher type q constant is at most T, where p<q≤2, the distortion is at most C(p,q,μ)T, independently of d. We also give explicit bounds as μ↓1. This answers the question of Johnson and Schechtman concerning the extension of their very tight embedding theorem below exponent one. The proof combines compression at a lower exponent with a change of density that preserves the number of coordinates, and applies over both scalar fields.
ER -