Hi,
the answer is: maybe?!
The problem is that obtaining the projected representation of your vertices might be hard and takes long, and may not be worth it. And besides that the output might be very ugly (very BIG rational numbers, or not even rational). So it may be that looking at the projected coordinates make things worse. But on the other hand the matrices you are working with are smaller so there is a chance that computational time does indeed get better, but I suspect only by a small margin.
But my warning from before is still valid. The Problem is that the number of possible facets gets really really big. So that even listing these wouldn't work. So 5 hours is not very surprising or even considered long in your context. When you don't know anything about the structure of your polytope, the odds are not in your favour. When something finishes in this big input size, you can consider yourself lucky
But there are some pointers I can give you:
- exploid every symmetry you know to reduce the input size
- try different algorithms with the polymake command: prefer "...", where you might use: cdd, lrs, beneath_beyond or ppl. See: http://arxiv.org/abs/1408.4653
- the program lrs has an estimate function (check the lrs manual), where it gives you an estimate on how many facets your polytope has. You might want to run this estimate a couple of times since apperantly it has a high variance.
As for your 'trivia' question:
I think the longest I witnessed for one convex hull computation to finish was ~10 hours over night. But from other people I heared that they waited for days or weeks for a convex hull algorithm to finish until they stopped the computation.