  Generalized metrics, arising from Lawvere's view of metric spaces as enriched
categories, have been widely applied in denotational semantics as a way to
measure to which extent two programs behave in a similar, although non
equivalent, way. However, the application of generalized metrics to
higher-order languages like the simply typed lambda calculus has so far proved
unsatisfactory. In this paper we investigate a new approach to the construction
of cartesian closed categories of generalized metric spaces. Our starting point
is a quantitative semantics based on a generalization of usual logical
relations. Within this setting, we show that several families of generalized
metrics provide ways to extend the Euclidean metric to all higher-order types.
