  Persistence diagrams are common objects in the field of Topological Data
Analysis. They are topological summaries that capture both topological and
geometric structure within data. Recently there has been a surge of interest in
developing tools to statistically analyse populations of persistence diagrams,
a process hampered by the complicated geometry of the space of persistence
diagrams. In this paper we study the median of a set of diagrams, defined as
the minimizer of an appropriate cost function analogous to the sum of distances
used for samples of real numbers. We then characterize the local minima of this
cost function and in doing so characterize the median. We also do some
comparative analysis of the properties of the median and the mean.
