The most important problem in extreme-value statistics is the question estimation of a quantile connected with a very low probability (for example estimating the height of a sea dike corresponding to a given failure probability). Since both parametric and non-parametric contexts are inappropriate (the parametric context being too narrow and the non-parametric context not offering real solutions), the semi-parametric context of extreme-value theory seems natural: the conditions are widely applicable, yet the conditions are sufficiently precise in order to provide realistic solutions. A similar problem comes up in
higher-dimensional space (involving e.g. sea levels and wave heights). It has completely new features. We propose to continue the development of relevant methods for solving this problem in finite-dimensional space and initiate the infinite-dimensional case which is also relevant for applications.