The k principal points [xi]1, ..., [xi]k of a random vector X are the points that approximate the distribution of X by minimizing the expected squared distance of X to the nearest of the [xi]j. A given set of k points y1, ..., yk partition p into domains of attraction D1, ..., Dk respectively, where Dj,consists of all points x [set membership, variant] p such that [short parallel]x - yj[short parallel] < [short parallel]x - yl[short parallel], l [not equal to] j. If E[X X [set membership, variant] Dj] = yj for each j, then y1, ..., yk are k self-consistent points of X ([short parallel]·[short parallel] is the Euclidian norm). Principal points are a special case of self-consistent points. Principal points and sell-consistent points are cluster means of a distribution and represent a generalization of the population mean from one to several points. Principal points and self-consistent points are studied for a class of strongly symmetric multivariate distributions. A distribution is strongly symmetric if the distribution of the principal components (Z1, ..., Zp)' is invariant up to sign changes, i.e., (Z1, ..., Zp)' has the same distribution as (±Z1, ..., ±Zp)'. Elliptical distributions belong to the class of strongly symmetric distributions. Several results are given for principal points and self-consistent points of strongly symmetric multivariate distributions. One result relates self-consistent points to principal component subspaces. Another result provides a sufficient condition for any set of self-consistent points lying on a line to be symmetric to the mean of the distribution.
