It is known that a bivariate extreme value distribution (EVD) \(G\) with reverse exponential margins can be represented as \(G(x,y)=\exp(-||(x,y)||)\), \(x,y\le 0\), where \(||\cdot||\) is a suitable norm on \(\mathbb{R}^2\). We prove in this paper the converse implication, i.e., given an arbitrary norm \(||\cdot||\) on \(\mathbb{R}^2\), \(G(x,y):=\exp(-||(x,y)||)\), \(x,y\le 0\), defines an EVD with reverse exponential margins, if and only if the norm satisfies for \(z\in[0,1]\) the condition \(\max(z,1-z)\le ||(z,1-z)||\le 1\). This result is extended to bivariate EVDs with arbitrary margins as well as to extreme value copulas. By identifying an EVD \(G(x,y)=\exp(-||(x,y)||)\), \(x,y\le 0\), with the unit ball corresponding to the generating norm \(||\cdot||\), we obtain a characterization of the class of EVDs \(G\) in terms of compact and convex subsets of \(\mathbb{R}^2\).