  We propose center-outward superquantile and expected shortfall functions,
with applications to multivariate risk measurements, extending the standard
notion of value at risk and conditional value at risk from the real line to
$\mathbb{R}^d$. Our new concepts are built upon the recent definition of
Monge-Kantorovich quantiles based on the theory of optimal transport, and they
provide a natural way to characterize multivariate tail probabilities and
central areas of point clouds. They preserve the univariate interpretation of a
typical observation that lies beyond or ahead a quantile, but in a meaningful
multivariate way. We show that they characterize random vectors and their
convergence in distribution, which underlines their importance. Our new
concepts are illustrated on both simulated and real datasets.
