  We bound the variance and other moments of a random vector based on the range
of its realizations, thus generalizing inequalities of Popoviciu (1935) and
Bhatia and Davis (2000) concerning measures on the line to several dimensions.
This is done using convex duality and (infinite-dimensional) linear
programming.
  The following consequence of our bounds exhibits symmetry breaking, provides
a new proof of Jung's theorem (1901), and turns out to have applications to the
aggregation dynamics modelling attractive-repulsive interactions: among
probability measures on ${\mathbf R}^n$ whose support has diameter at most
$\sqrt{2}$, we show that the variance around the mean is maximized precisely by
those measures which assign mass $1/(n+1)$ to each vertex of a standard
simplex.
  For $1 \le p <\infty$, the $p$-th moment --- optimally centered --- is
maximized by the same measures among those satisfying the diameter constraint.
