Isometric embeddings between classical Banach spaces, cubature formulas, and spherical designs (Q685203)

The authors prove that if an isometric embedding \(\ell^ m_ p \to \ell^ n_ q\) with finite \(p,q>1\) exists, then \(p=2\) and \(q\) is an even integer. They denote by \(N(m,q)\) the least number \(n\) for which there exists an isometric embedding \(\ell^ m_ 2 \to \ell^ n_ q\) and prove that \[ {m+q/2-1 \choose m-1} \leq N(m,q) \leq {m+q-1 \choose m-1}. \] Using approach based on relation between embeddings, Euclidean spherical designs and cubature formulas the authors sharpen the above lower bound for \(N(m,q)\) and obtain a series of concrete values, e.g. \(N(3,8)=16\). At the end of the paper the authors consider the problem of existence of \(\varepsilon\)-isometric embeddings of \(\ell^ 3_ 2\).