Isometric embedding of the n-point spaces into the space of spaces for $n \leq 4$

Abstract

In [The Space of Spaces: Curvature Bounds and Gradient Flows on the Space of Metric Measure Spaces. Memoirs of the American Mathematical Society. American Mathematical Society, 2023], Sturm studied the space of all metric measure spaces up to isomorphism which he called The space of spaces. He also introduced for a natural number n the space of all n-points metric spaces. The aim of this article is to study if the embedding of this space in the space of spaces is isometric. Using results from [Haggai Maron and Yaron Lipman. (probably) concave graph matching. Advances in Neural Information Processing Systems, 31, 2018] and [Hiroshi Maehara. Euclidean embeddings of finite metric spaces. Discrete Mathematics, 2013], we prove that it is the case for $n \leq 4$ and for Euclidean metric spaces.