I think it's impossible in general. Try with the disk ; all you can get by taking convex combinations of the extreme points is a polygon inscribed inside it, you can never get the whole set. And I think (let's say I... leave this as an exercise =P) that the disk is convex.
In fact, my intuition tells me that this is only possible with polyhedrons ; as soon as you have little curve in there, the curve cannot be "polygonized" with extreme points... and with polyhedrons, well.
If it was clear that you were trying to do that with polyhedrons then perhaps this "n poly(n)" notation was kind of weird.
Since it is easy to deal with convex combinations inside triangles (or n dimensional tetrahedrons, because a basis is easily found) I guess you could try to divide your polyhedron into "sections" by choosing one arbitrary point and letting a straight line link it to all other points. Therefore your convex polyhedron will be divided in triangles (or n-tetrahedrons) in which convex combinations are easily taken with linear algebra.
Hope that helps,