Hoop conjecture for apparent horizon formation

We search for apparent horizons in various kinds of analytic initial data with axisymmetry but which allow some extreme matter configurations, in order to gain some insights into the hoop conjecture proposed by Thorne. The problem of defining the circumference C of a body is also considered, and a precise definition of C is proposed. For the models considered, the necessary condition for the apparent horizon formation is found to be where M is the ADM mass. Although initial data investigated here are only of a restricted class, it is interesting to note that this criterion almost coincides with the result obtained by Barrabès et al and by Tod who investiged a convex null dust shell.