  An $n\times n$ symmetric matrix $A$ is copositive if the quadratic form
$x^TAx$ is nonnegative on the nonnegative orthant. The cone of copositive
matrices strictly contains the cone of completely positive matrices, i.e., all
matrices of the form $BB^T$ for some (possibly rectangular) matrix $B$ with
nonnegative entries. The main result, proved using Blekherman's real algebraic
geometry inspired techniques and tools of convex geometry, shows that
asymptotically, as $n$ goes to infinity, the ratio of volume radii of the two
cones is strictly positive. Consequently, the same holds true for the ratio of
volume radii of any two cones sandwiched between them, e.g., the cones of
positive semidefinite matrices, matrices with nonnegative entries, their
intersection and their Minkowski sum.
