  In this paper, we present a geometric condition for a family of CAT(0)
spaces, which ensures that the Izeki-Nayatani invariants of spaces in the
family are uniformly bounded from above by a constant strictly less than 1.
Each element of such a family with this condition is a space presented by M.
Gromov as an example of a "CAT(0) space with "bounded" singularities".
Combining our result with a result of Izeki, Kondo and Nayatani, we see that
random groups of Gromov's graph model have a fixed-point property for such a
family.
