  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.
