Is it possible to cover the cone of positive semidefinite matrices by a finite/countable/interesting family of closed intervals of matrices?
How about a general convex cone?
For the finite case the answer seems to be no but maybe there is some ingenious way I am missing.
EDIT: What I mean by interval $[A,B]$ is the set of convex combinations of the matrices $A$ and $B$. Other definitions of interval are possible, for example taking "convex combinations" with the weight scalars replaced by diagonal matrices. (like in this paper - http://www.math.wsu.edu/faculty/tsat/files/jt.pdf). Or ot can be defined entrywise, woth all matrices in the interval being entrywise greater than $A$ and less than $B$. Hope it's clearer now.