  The study of multivariate extremes is dominated by multivariate regular
variation, although it is well known that this approach does not provide
adequate distinction between random vectors whose components are not always
simultaneously large. Various alternative dependence measures and
representations have been proposed, with the most well-known being hidden
regular variation and the conditional extreme value model. These varying
depictions of extremal dependence arise through consideration of different
parts of the multivariate domain, and particularly exploring what happens when
extremes of one variable may grow at different rates to other variables. Thus
far, these alternative representations have come from distinct sources and
links between them are limited. In this work we elucidate many of the relevant
connections through a geometrical approach. In particular, the shape of the
limit set of scaled sample clouds in light-tailed margins is shown to provide a
description of several different extremal dependence representations.
