  The surface density of the star formation rate in different galaxies, as well
as in different parts of a single galaxy, scales nonlinearly with the surface
density of the total gas. This observationally established relation is known as
the Kennicutt-Schmidt star formation law. The slope of the star formation law
has been shown to change with the density of the gas against which the star
formation rate is plotted. This dependence implies a nonlinear scaling between
the dense gas and the total gas surface densities within galaxies. Here, we
explore a possible interpretation of this scaling as a property of the geometry
of the interstellar medium (ISM), and we find that it arises naturally if the
topology of the ISM is multifractal. Under the additional assumption that, at
very high densities, the star formation timescale is roughly constant, the star
formation law itself can also be recovered as a consequence of the multifractal
geometry of the ISM. The slope of the scaling depends on the width of the
global probability density function (PDF), and is between 1.5 and 1.6 for wide
PDFs relevant to high-mass systems, while it is higher for narrower PDFs
appropriate for lower-mass dwarf galaxies, in agreement with observations.
