  Spherical symmetry arguments are used to produce a general device to convert
identities and inequalities for the $p$th absolute moments of real-valued
random variables into the corresponding identities and inequalities for the
$p$th moments of the norms of random vectors in Hilbert spaces. Particular
results include the following: (i) an expression of the $p$th moment of the
norm of such a random vector $X$ in terms of the characteristic functional of
$X$; (ii) an extension of a previously obtained von~Bahr--Esseen-type
inequality for real-valued random variables with the best possible constant
factor to random vectors in Hilbert spaces, still with the best possible
constant factor; (iii) an extension of a previously obtained inequality between
measures of "contrast between populations" and "spread within populations" to
random vectors in Hilbert spaces.
