  This paper will introduce a family of sliced Wasserstein geodesics which are
not standard Wasserstein geodesics, objects yet to be discovered in the
literature. These objects exhibit how the geometric structure of the Sliced
Wasserstein space differs from the Wasserstein space, and provides a simple
example of how solving the barycenter and gradient flow problems change when
moving between these metrics. Some of these geodesics will only be H\"older
continuous with respect to the Wasserstein metric and thus will provide a
direct proof that Sliced-Wasserstein and regular Wasserstein metrics are not
equivalent. Previous proofs of this were done for various cases in [2] and [5].
This paper, not only provides a direct proof, but also fills in gaps showing
these metrics not equivalent in dimensions greater than 2.
