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

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 4 and for Euclidean metric spaces.