Intrinsic Isometric Embeddings of Pro-Euclidean Spaces

Petrunin proves that a metric space $\mathcal{X}$ admits an intrinsic isometry into $\mathbb{E}^n$ if and only if $\mathcal{X}$ is a pro-Euclidean space of rank at most $n$. He then shows that either case implies that $\mathcal{X}$ has covering dimension $\leq \, n$. In this paper we extend this result to include embeddings. Namely, we first prove that any pro-Euclidean space of rank at most $n$ admits an intrinsic isometric embedding into $\mathbb{E}^{2n+1}$. We then discuss how Petrunin's result implies a partial converse to this result.