  It is known that for a variety of choices of metrics, including the standard
bottleneck distance, the space of persistence diagrams admits geodesics.
Typically these existence results produce geodesics that have the form of a
convex combination. More specifically, given two persistence diagrams and a
choice of metric, one obtains a bijection realizing the distance between the
diagrams, and uses this bijection to linearly interpolate from one diagram to
another. We prove that for several families of metrics, every geodesic in
persistence diagram space arises as such a convex combination. For certain
other choices of metrics, we explicitly construct infinite families of
geodesics that cannot have this form.
