  We develop a test for spherical symmetry of a multivariate distribution $P$
that works even when the dimension of the data $d$ is larger than the sample
size $n$. We propose a non-negative measure $\zeta(P)$ such that $\zeta(P)=0$
if and only if $P$ is spherically symmetric. We construct a consistent
estimator of $\zeta(P)$ using the data augmentation method and investigate its
large sample properties. The proposed test based on this estimator is
calibrated using a novel resampling algorithm. Our test controls the Type-I
error, and it is consistent against general alternatives. We also study its
behaviour for a sequence of alternatives $(1-\delta_n) F+\delta_n G$, where
$\zeta(G)=0$ but $\zeta(F)>0$, and $\delta_n \in [0,1]$. When
$\lim\sup\delta_n<1$, for any $G$, the power of our test converges to unity as
$n$ increases. However, if $\lim\sup\delta_n=1$, the asymptotic power of our
test depends on $\lim n(1-\delta_n)^2$. We establish this by proving the
minimax rate optimality of our test over a suitable class of alternatives and
showing that it is Pitman efficient when $\lim n(1-\delta_n)^2>0$. Moreover,
our test is provably consistent for high-dimensional data even when $d$ is
larger than $n$. Our numerical results amply demonstrate the superiority of the
proposed test over some state-of-the-art methods.
