Hold on a second. Did you mean that the best dissection involved the shortest fragment being coming from an equal size division of some stick? Or that the best dissection involved SOME stick being equally divided? Because I wasn't sure there was an obvious reason for the first, but the second could still be true (if I've got the right solution for 4<->7)?
Or rather, what happens if you try to increase all the shortest lengths by epsilon? That obviously fails if several of them compose a single stick. That's what you were trying to explain to me before. But if you follow through adjusting all other stick lengths epsilon as you described, it has to fail *somewhere* or you didn't start with a local optimum. Another way it could fail is if the *complements* of the shortest fragments are equal divisions of a stick. But there may be other ways as well, if you end up with a stick made of fragments that all have to get longer or all have to get shorter, but came from a different number of steps from the original perturbation.
But if the optimal solution for some division is 5/3, I can't help but wonder if that means *something* has to be divided into three equal pieces.
I wonder if this might be easier to imagine looking at perturbing a matrix as you described, instead of sticks. |