Abstract

We settle some open problems regarding RPI-norms, i.e. norms (on a convenient subspace of differentiable real functions) that are invariant under differentiable reparametrizations. More precisely, we clarify some key points on how RPI-norms can be approximated using standard RPI-norms. We first prove that the sum of two standard RPI-norms is the supremum of a set of standard RPI-norms. This allows us to study a class of RPI-norms that can be expressed as the sup of standard ones. Finally, we give a positive answer to an open question on the existence of RPI-norms that are not the sup of standard RPI-norms.