The asymptotic efficiency, ARE_{p,2}, of the tests for multivariate means theta in \R^d based on the p-means relative to the standard 2-mean, (approximate) likelihood ratio test (LRT), is considered for large dimensions d. It turns out that these p-mean tests for p>2 may greatly outperform the LRT while never being significantly worse than the LRT. For instance, ARE_{p,2} for p=3 varies from about 0.96 to \infty, depending on the direction of the alternative mean vector theta_1, for the null hypothesis H_0: theta=\0. These results are based on a complete characterization, under certain general and natural conditions, of the varying pairs (n,theta_1) for which the values of the power of the p-mean test for theta=\0 and theta=theta_1 tend, respectively, to prescribed values alpha and beta. The proofs use such classic results as the Berry-Esseen bound in the central limit theorem and the conditions of convergence to a given infinitely divisible distribution, as well as a recent result by the author on the Schur^2-concavity properties of Gaussian measures.