Gang Cai

Very tight embeddings of subspaces of \(L_p\) for \(0<p<1\)

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.