A note on “Maximum distributions for l2,p-symmetric vectors are skewed l1,p-symmetric distributions” by Batún-Cutz et al. (2013)

In a recent paper,  Batún-Cutz et al. (2013) showed that the density of the maximum of the components of a l2,p-symmetrically distributed random vector have skewed l1,p-symmetric distribution. Their proof, based on a geometric measure representation for the distribution of the maximum, is quite involved and long. Here, we present a very simple proof of their main result by using a property of the distribution of the maximum of an exchangeable bivariate random vector.