  Context: The gas density structure of the cold molecular phase of the
interstellar medium is the main controller of star formation. Aims: A
theoretical framework is proposed to describe the structural content of the
density field in isothermal supersonic turbulence. Methods: It makes use of
correlation and structure functions of the phase indicator field defined for
different iso-density values. The relations between these two-point statistics
and the geometrical features of iso-density sets such as the volume fraction,
the surface density, the curvature, and fractal characteristics are provided.
An exact scale-by-scale budget equation is further derived revealing the role
of the turbulent cascade and dilation on the structural evolution of the
density field. Although applicable to many flow situations, this tool is here
first invoked for characterising supersonic isothermal turbulence, using data
from the currently best-resolved numerical simulation. Results: We show that
iso-density sets are surface fractals rather than mass fractals, with
dimensions that markedly differ between dilute, neutral, and dense regions. The
surface-size relation is established for different iso-density values. We
further find that the turbulent cascade of iso-density sets is directed from
large towards smaller scales, in agreement with the classical picture that
turbulence acts to concentrate more surface into smaller volumes. Intriguingly,
there is no range of scales that complies with a constant transfer rate in the
cascade, challenging our fundamental understanding of interstellar turbulence.
Finally, we recast the virial theorem in a new formulation drawing an explicit
relation between the aforementioned geometrical measures and the dynamics of
iso-density sets.
