Projective metrics and contraction principles for complex cones
  In this article, we consider linearly convex complex cones in complex Banach spaces and we define a new projective metric on these cones. Compared to the hyperbolic gauge of Rugh, it has the advantage of being explicit, and easier to estimate. We prove that this metric also satisfies a contraction principle like Birkhoff's theorem for the Hilbert metric. We are thus able to improve existing results on spectral gaps for complex matrices. Finally, we compare the contraction principles for the hyperbolic gauge and our metric on particular cones, including complexification of Birkhoff cones. It appears that the contraction principles for our metric and the hyperbolic gauge occur simultaneously on these cones. However, we get better contraction rates with our metric. 
Loic Dubois