The multi-cover persistence of Euclidean balls

Given a locally finite set X in R^d and a positive radius, the k-fold cover of X and r consists of all points that have k or more points of X within distance r. The order-k Voronoi diagram decomposes the k-fold cover into convex regions, and we use the nerve of this decomposition to compute homology and persistence. In particular the persistence in depth is interesting from a geometric as well as algorithmic viewpoint. The main tool in understanding its structure is a rhomboid tiling in R^{d+1} that combines the nerves for all values of k into one. We mention a straightforward consequence, namely that the cells in the nerve are generically not simplicial, unless k=1 or d=1,2.

(Joint work with Georg Osang.)