  Shibata and Sasaki [arXiv:gr-qc/9905064] introduced the so-called compaction
function. Since then, it has been empirically established that the maximum
value of this function (or its volume-averaged counterpart) in the
long-wavelength solutions gives a very robust threshold of primordial black
hole formation. In this paper, we show that in spite of initial intention, the
Shibata-Sasaki compaction function cannot be interpreted as the ratio of the
mass excess to the areal radius in the constant-mean-curvature slice of their
choice but coincides with that in the {\it comoving} slice up to a constant
factor depending on the equation of state. We also discuss the
gauge-(in)dependence of the legitimate compaction function, i.e., the ratio of
the mass excess to the areal radius, in the long-wavelength solutions.
