On a Characteristic Property of the Normal Law

As it is well known the normal distribution is characterized by the uniformity of the distribution of the random vector $\biggl(\dfrac{X_1-\bar X}s,\dots,\dfrac{X_n-\bar X}s\biggr)$ on the unit sphere (here we use usual notations). It is shown that there exists a set of triplets of points of that sphere such that the normality of the sample follows from the constancy of the density of that vector only on any one of these triplets.