Structure in the 3D Galaxy Distribution: II. Voids and Watersheds of
  Local Maxima and Minima
  The major uncertainties in studies of the multi-scale structure of the
Universe arise not from observational errors but from the variety of legitimate
definitions and detection methods for individual structures. To facilitate the
study of these methodological dependencies we have carried out 12 different
analyses defining structures in various ways. This has been done in a purely
geometrical way by utilizing the HOP algorithm as a unique parameter-free
method of assigning groups of galaxies to local density maxima or minima. From
three density estimation techniques (smoothing kernels, Bayesian Blocks and
self organizing maps) applied to three data sets (the Sloan Digital Sky Survey
Data Release 7, the Millennium Simulation and randomly distributed points) we
tabulate information that can be used to construct catalogs of structures
connected to local density maxima and minima. The resulting sizes follow
continuous multi-scale distributions with no indication of the presence of a
discrete hierarchy. We also introduce a novel void finder that utilizes a
method to assemble Delaunay tetrahedra into connected structures and
characterizes regions very nearly empty of galaxies in the source catalog.
