Covering the cone of positive semidefinite matrices by intervals 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.  
 A: *

*The definition of an "interval" given in the current edit trivially implies that every interval has empty interior in the cone of PSD matrices. The same holds for any other convex cone of dimension at least 2. Therefore, by the Baire category theorem, a countable union of such intervals cannot cover the cone. 

*The alternative definition given in the edit, using entry-wise comparison of matrices, gives you "intervals" (I will call them "boxes") with nonempty interior. Then, trivially, there is countable cover of the cone by boxes (even a locally finite one, since the space is paracompact). I do not know if such a cover is "interesting", it depends on what you are interested in. You may want to define first the notion of an "interesting" cover of an open disk by rectangular boxes. On the other hand, since every closed box is compact, there is no finite covering (as the cone in question is noncompact).  

*With Yemon's interpretation, the answer is still  the same as in (2), but one needs to do a bit more work. (Note that every convex cone $C$ in $R^n$ determines two partial orders on $R^n$: One where $C$ is positive and the other where the dual cone is positive, both definitions are natural.)   
