Is there a general name for densities of the following form? $$ p(x) \propto \exp(-\|x\|_r^s), \quad r \in [1, \infty], s \in [1, \infty). $$ For $r=s=2$, it is a product of Gaussians, and for $r=s=1$, it is a product of Laplace distributions.
I'm particularly interested in $r = \infty, s = 1$, but the general case above is interesting. It is not even clear to me that this is always a density (i.e., normalizable).
I can at least tell you that its always a density for your range of indices. By equivalence of norms, one has $$ \exp(-\|x\|_r^s) \le \exp(-C^s_{r,d}\|x\|^s_1 ) \le \exp(-C^s_{r,d}\|x\|_1 )$$ which shows that it has a finite integral, which is of course the normalising constant.