The main results imply that the probability P(\ZZ\in A+þ) is Schur-concave/Schur-convex in (þ_1^2,\dots,þ_k^2) provided that the indicator function of a set A in \R^k is so, respectively; here, þ=(þ_1,\dots,þ_k) in \R^k and \ZZ is a standard normal random vector in \R^k. Moreover, it is shown that the Schur-concavity/Schur-convexity is strict unless the set A is equivalent to a spherically symmetric set. Applications to testing hypotheses on multivariate means are given.