I would like to characterise the subspaces of $ \ell_p^n(\mathbb{R})$ that are isometric (for $ p$ an even integer). In the literature, I have found few results related to this.

One can show that $ A \in \mathbb{R}^{m \times n}$ is an isometry from $ \ell_p^n(\mathbb{R}^n) \to \ell_p^n(\mathbb{R}^m)$ if and only if the columns of $ A$ have unit $ \ell_p$ -norm and only one nonzero entry per column. Taking $ n \le m$ , this characterises the subspaces isometric to the subspace $ \ell_p^n(\mathbb{R}^n) \subset \ell_p^n(\mathbb{R}^m)$ .

Also, for $ p$ not an even integer, I believe all the isometries between subspaces of $ L_p([0,1])$ have been classified, and this can likely be extended to $ \ell_p^n(\mathbb{R})$ .

Does anyone know any references in this direction for even $ p$ ?