  The original spin foam model construction for 4D gravity by Barrett and Crane
suffers from a few troubling issues. In the simple examples of the vertex
amplitude they can be summarized as the existence of contributions to the
asymptotics from non geometric configurations. Even restricted to geometric
contributions the amplitude is not completely worked out. While the phase is
known to be the Regge action, the so called measure factor has remained
mysterious for a decade. In the toy model case of the 6j symbol this measure
factor has a nice geometric interpretation of $V^{-1/2}$ leading to
speculations that a similar interpretation should be possible also in the 4D
case. In this paper we provide the first geometric interpretation of the
geometric part of the asymptotic for the spin foam consisting of two glued
4-simplices (decomposition of the 4-sphere) in the Barrett-Crane model in the
large internal spin regime.
