  The Hilbert metric on convex subsets of $\mathbb R^n$ has proven a rich
notion and has been extensively studied. We propose here a generalization of
this metric to subset of complex projective spaces and give examples of
applications to diverse fields. Basic examples include the classical Hilbert
metric which coincides with the hyperbolic metric on real hyperbolic spaces as
well as the complex hyperbolic metric on complex hyperbolic spaces.
