  Massive neutrinos suppress the growth of cosmic structure on nonlinear
scales, motivating the use of information beyond the power spectrum to tighten
constraints on the neutrino mass, for example by considering cosmic voids. It
was recently proposed that constraints on neutrino mass from the halo mass
function (HMF) can be improved by considering only the halos that reside within
voids -- the void-halo mass function (VHMF). We extend this analysis, which
made spherical assumptions about the shape of voids, to take into account the
non-spherical nature of voids as defined by the Voronoi-tessellation-based void
finder, VIDE. In turn, after accounting for one spurious non-spherical void, we
find no evidence that the VHMF contains information beyond the HMF. Given this
finding, we then introduce a novel summary statistic by splitting halos
according to the emptiness of their individual environments, defined by the
Voronoi cell volume each halo resides in, and combining the mass functions from
each split. We name the corresponding statistic the VorHMF and find that it
could provide information regarding neutrino mass beyond the HMF. Our work thus
motivates the importance of accounting for the full shape of voids in future
analyses, both in terms of removing outliers to achieve robust results and as
an additional source of cosmological information.
