  Let $K$ be a closed convex polyhedron defined by a finite number of linear
inequalities. In this paper we refine the theory of abstract tubes (Naiman and
Wynn, 1997) associated with $K$ when $K$ is perturbed. In particular, we focus
on the perturbation that is lexicographic and in an outer direction. An
algorithm for constructing the abstract tube by means of linear programming and
its implementation are discussed. Using the abstract tube for perturbed $K$
combined with the recursive integration technique proposed by Miwa, Hayter and
Kuriki (2003), we show that the multidimensional normal probability for a
polyhedral region $K$ can be computed efficiently. In addition, abstract tubes
and the distribution functions of studentized range statistics are exhibited as
numerical examples.
