the problem is that you want to cut up an onion in such a way as to minimize variation in the size and shape of the cut-up pieces
usually, so that the pieces will cook evenly
I don't do much with my food processor anymore besides grating cheese; even biscuit dough I'll do with a box grater at this point, just to avoid having to clean out the food processor.
I agree that cleaning the food processor is more of a pain than cleaning the knife. On the other hand, using it is far less of a pain than using the knife (especially in cases like this where you're trying to get even, small pieces). So you're really trading off one pain for another. It's not clear to me that either option is the obvious winner or loser here.
If you have to clean things by hand, I'd take the awkward inside of a smooth plastic cylinder over a grater any day.
It's not impossible to clean and worth it when making larger amounts but definitely more of a hassle than a knife, even with a dish washer.
It's more of a geometry thought experiment than a practical epicurean "problem".
Not very well. There are some snippets:
"to keep the pieces as similar as possible"
"The Jacobian r dr dθ gives a measure of how big the infinitely small pieces are relative to each other"
"The variance is a good measure of the uniformity of the pieces."