On Computing a Center Persistence Diagram

Throughout this paper, a persistence diagram ${cal P}$ is composed of a set $P$ of planar points (each corresponding to a topological feature) above the line $Y=X$, as well as the line $Y=X$ itself, i.e., ${cal P}=Pcup{(x,y)|y=x}$. Given a set of persistence diagrams ${cal P}_1,...,{cal P}_m$, for the data reduction purpose, one way to summarize their topological features is to compute the {em center} ${cal C}$ of them first under the bottleneck distance. We consider two discre