  We use a functional analogue of the quantile function for probability
measures admitting a continuous Lebesgue density on $\mathbb{R}^d$ to
characterise the class of non-trivial limit distributions of radially
recentered and rescaled multivariate exceedances. A new class of multivariate
distributions is identified, termed radially-stable generalised Pareto
distributions, and is shown to admit certain stability properties that permit
extrapolation to extremal sets along any direction in cones such as
$\mathbb{R}^d$ and $\mathbb{R}_+^d$. Leveraging the limit Poisson point process
likelihood of the point process of radially renormalised exceedances, we
develop parsimonious statistical models that exploit theoretical links between
structural star-bodies and are amenable to Bayesian inference. Our framework
sharpens statistical inference by suitably including additional information
from the angular directions of the geometric exceedances and facilitates
efficient computations in dimensions $d=2$ and $d=3$. Additionally, it
naturally leads to the notion of return level-set, which is a canonical
quantile set expressed in terms of its average recurrence interval, and a
geometric analogue of the uni-dimensional return level. We illustrate our
methods with a simulation study showing superior predictive performance of
probabilities of rare events, and with two case studies, one associated with
river flow extremes, and the other with oceanographic extremes.
