  The set of trace preserving, positive maps acting on density matrices of size
d forms a convex body. We investigate its nested subsets consisting of
k-positive maps, where k=2,...,d. Working with the measure induced by the
Hilbert-Schmidt distance we derive asymptotically tight bounds for the volumes
of these sets. Our results strongly suggest that the inner set of
(k+1)-positive maps forms a small fraction of the outer set of k-positive maps.
These results are related to analogous bounds for the relative volume of the
sets of k-entangled states describing a bipartite d X d system.
