We describe the Wasserstein space for the homogeneous H1 metric which turns out to be isometric to (a piece of) an infinite-dimensional sphere. The corresponding geodesic flow turns out to be integrable, and it is a generalization of the Hunter-Saxton equation. The corresponding optimal transport can be used for the ßize-recognition", as opposed to the ßhape recognition". This is a joint work with J. Lenells, G. Misiolek, and S. Preston.