A scalar product for copulas

We introduce a scalar product for n-dimensional copulas, based on the Sobolev scalar product for W1,2-functions. The corresponding norm has quite remarkable properties and provides a new, geometric framework for copulas. We show that, in the bivariate case, it measures invertibility properties of copulas with respect to the ∗-operation introduced by Darsow et al. (1992). The unique copula of minimal norm is the null element for the ∗-operation, whereas the copulas of maximal norm are precisely the invertible elements.