  We show that any distribution function on $\mathbb{R}^d$ with nonnegative,
nonzero and integrable marginal distributions can be characterized by a norm on
$\mathbb{R}^{d+1}$, called $F$-norm. We characterize the set of $F$-norms and
prove that pointwise convergence of a sequence of $F$-norms to an $F$-norm is
equivalent to convergence of the pertaining distribution functions in the
Wasserstein metric. On the statistical side, an $F$-norm can easily be
estimated by an empirical $F$-norm, whose consistency and weak convergence we
establish.
  The concept of $F$-norms can be extended to arbitrary random vectors under
suitable integrability conditions fulfilled by, for instance, normal
distributions. The set of $F$-norms is endowed with a semigroup operation
which, in this context, corresponds to ordinary convolution of the underlying
distributions. Limiting results such as the central limit theorem can then be
formulated in terms of pointwise convergence of products of $F$-norms.
  We conclude by showing how, using the geometry of $F$-norms, we may
characterize nonnegative integrable distributions in $\mathbb{R}^d$ by simple
compact sets in $\mathbb{R}^{d+1}$. We then relate convergence of those
distributions in the Wasserstein metric to convergence of these characteristic
sets with respect to Hausdorff distances.
