Let be i.i.d. points uniformly on the unit sphere in , , and let be the random polyhedron generated by . Furthermore, for linearly independent vectors u in , let be the number of shadow vertices of X in span . The paper provides an asymptotic expansion of the expectation value for fixed n and , equals the expected number of pivot steps that the shadow vertex algorithm - a parametric variant of the simplex algorithm - requires in order to solve linear programming problems of type , , if the algorithm will be started with an X-vertex solving the problem, , . The present analysis is closely related to Borgward's probabilistic analysis of the simplex algorithm. The paper obtains a refined asymptotic analysis of the expected number of pivot steps required by the shadow vertex algorithm for uniformly on the sphere distributed data.
