  While max-stable processes are typically written as pointwise maxima over an
infinite number of stochastic processes, in this paper, we consider a family of
representations based on $\ell^p$ norms. This family includes both the
construction of the Reich-Shaby model and the classical spectral representation
by de Haan as special cases. As the representation of a max-stable process is
not unique, we present formulae to switch between different equivalent
representations. We further provide a necessary and sufficient condition for
the existence of a $\ell^p$ norm based representation in terms of the stable
tail dependence function of a max-stable process. Finally, we discuss several
properties of the represented processes such as ergodicity or mixing.
