Upper bound for isometric embeddings \ell_2^m\to\ell_p^n

TL;DR

This paper establishes an upper bound on the minimal dimension needed for isometric embeddings from ^m to ^n over various fields, extending previous results in the non-Hilbert case.

Contribution

It provides a new upper bound for the minimal n in isometric embeddings ^m to ^n over real, complex, and quaternionic fields, generalizing earlier work.

Findings

Derived an explicit upper bound for n in isometric embeddings.

Extended previous bounds to quaternionic and complex fields.

Confirmed the bound matches known results in the real case.

Abstract

The isometric embeddings $\ell_{2;K}^m\to\ell_{p;K}^n$ ($m\geq 2$, $p\in 2\N$) over a field $K\in{R, C, H}$ are considered, and an upper bound for the minimal $n$ is proved. In the commutative case ($K\neq H$) the bound was obtained by Delbaen, Jarchow and Pe{\l}czy{\'n}ski (1998) in a different way.