Summary: Suppose that an order restriction is imposed among several \(p\)-variate normal mean vectors. We are interested in the problems of estimating these mean vectors and testing their homogeneity under this restriction. These problems are multivariate extensions of D. J. Bartholomew’s [Biometrika 46, 36-48 (1959; Zbl 0087.14202)] ones. For the bivariate case, these problems have been studied by S. Sasabuchi et al. [see Am. J. Math. Manage Sci. 18, No. 1-2, 131-158 (1998; Zbl 0919.62055)] and some others. In the present paper we examine the convergence of an iterative algorithm for computing the maximum likelihood estimator when \(p\) is larger than two. We also study some test procedures for testing homogeneity when \(p\) is larger than two.