Intrinsic Isometric Embeddings of Pro-Euclidean Spaces

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Petrunin proves that a metric space X admits an intrinsic isometry into Eⁿ if and only if X is a pro-Euclidean space of rank at most n. He then shows that either case implies that X has covering dimension ≤ 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 E²ⁿ⁺¹. We then discuss how Petrunin's result implies a partial converse to this result.