Isometric rigidity of metric constructions with respect to Wasserstein spaces

In this paper we study the isometric rigidity of certain classes of metric spaces with respect to the p-Wasserstein space.We prove that spaces that split a separable Hilbert space are not isometrically rigid with respect to P 2 .We then prove that geodesic rays are isometrically rigid with respect to P p for any p ≥ 1, whereas taking infinite half-cylinders (i.e.product spaces of the form X × [0, ∞)) over compact non-branching geodesic spaces preserves isometric rigidity with respect to P p , for p > 1.Finally, we prove that spherical suspensions over compact spaces with diameter at most π/2 are isometrically rigid with respect to P p , for p > 1.