### 3.7 Avoiding collapse

Starting from the observation that Regge configurations exist with and , Beirl et
al. [31, 32] investigated the influence of a cutoff on the fatness of a simplex , defined by
Such a uniform shrinking of simplices is known to be necessary in order for piecewise flat manifolds to
approach their continuum counterparts [82]. On the 4^{4}-lattice, with the -measure and ,
they determined for decreasing . For small , the choice of seems to have
only little influence, but towards the transition point , simultaneously decreases.
Next they investigated a variety of measures of the form (see also [35]). For ,
only a mild -dependence of is observed as . However, for , there
are significant differences for the entire range of , and some evidence that the geometry
degenerates.