||Wed 2014-02-19 14:01|
|Indeed, but I haven't yet found such a limit, or even a proof of rationality!|
As I was saying to someone in another forum:
One would like to think that there's some nice proof involving local perturbations – pick the smallest fragment size and adjust it by ε, then adjust some other fragment of the same stick by −ε to make up for it, which means in turn that a fragment of the appropriate stick at the other end of the dissection goes by +ε, and ... you hope to end up with a closed loop of adjustments, or several such loops superimposed, and hence an adjustment to your original dissection which is legal for some sufficiently small ε > 0 and increases the length of every currently minimal fragment. If you can do that, then the dissection you started with was not a local optimum, and hence not a global optimum either.
The point of doing this would be that it gives you a reason why the smallest fragment keeps turning out to be achieved by cutting a single stick into equal pieces – because then clearly no such perturbation can increase the length of all those pieces at once, or else the result wouldn't fit in that stick any more.
But I don't actually have a proof along those lines, only a sketch of the sort of thing I'd like the proof to achieve. And even that wouldn't prove that an optimum dissection had to have all fragments rational, only that the smallest fragment had to be rational.
In fact I'd guess it's quite likely that optimal dissections need not have the longer fragments all rational (there's probably some situation in which you can find a closed loop of those local perturbations that never touch a minimal fragment, in which case you can apply that change with irrational ε to get a partially irrational and equally good answer), but I'd at least hope that there exists a rational optimal dissection in every case...