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We consider Hotelling's T^2 statistic for an arbitrary d-dimensional sample.If the sampling is not too deterministic or inhomogeneous, then under zeromeans hypothesis, T^2 tends to \chi^2_d in distribution. We show that a testfor the orthant symmetry condition introduced by Efron can be constructed whichdoes not essentially differ from the one based on \chi^2_d and at the same timeis applicable not only for large random homogeneous samples but for allmultidimensional samples without exceptions. The main assertions have the formof inequalities, not that of limit theorems; these inequalities are exactrepresenting the solutions to certain extremal problems. Let us also mention anauxiliary result which itself may be of interest: \chi_d-(d-1)^{1/2} decreasesin distribution in d to its limit N(0,1/2).

Extremal Probabilistic Problems and Hotelling's $T^2$ Test Under a Symmetry Condition