Very tight embeddings of subspaces of $L_p$ for $0<p<1$

Gang Cai; Longfa Sun

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arXiv · 2610.30546

Very tight embeddings of subspaces of $L_p$ for $0<p<1$

Gang Cai·Longfa Sun

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Abstract

For $0 1$, every $d$-dimensional subspace of $L_p$ embeds linearly into $\ell_p^{\lfloorμd\rfloor}$ with distortion at most $C(p,μ)(\log(d+2))^{1/p}$. If its Rademacher type $q$ constant is at most $T$, where $p<q\leq2$, the distortion is at most $C(p,q,μ)T$, independently of $d$. We also give explicit bounds as $μ\downarrow1$. 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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Gang Cai, Longfa Sun. 2026-10-22. Very tight embeddings of subspaces of $L_p$ for $0<p<1$. https://arxiv.org/abs/2610.30546

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