  We introduce a notion of p-dimensional cones made of $p$-dimensional
subspaces and gauges on these cones, giving rise to a contraction principle
which generalizes the one for Birkhoff cones. Using tools on the grassmannian
and the exterior algebra, we prove a spectral gap result for the p largest
eigenvalues of an operator and a regularity result for the characteristic
exponents of a random product of cone-contracting operators.
