How often is a random quantum state k-entangled?

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 kpositive 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 × d system.