The problem is that the constant function $f(Y) = 4$ is always independent of $X$, so nothing can satisfy this definition (right?).

@PerAlexandersson, ha, that would be outside the usual model of preferences, which are only allowed to be measures on the cake.

I don't see the harm in letting the question stay open to see what happens, especially if the main danger is cool stuff.

In sum, it seems reasonable to assume knife movements must be continuous, or else it seems that we need to more precisely define the rules.

The necessary formalization does not seem in place to deal with your scenario. If adversary 1 yells cut at precisely the same moment that adversary 2 makes a discontinuous knife movement, what should happen? What if Marge and adversary 1 both have a strategy of "yell immediately after a discontinuous knife movement satisfying ____", who yells first or is it a tie? If it is a tie, I think you're right and the allocation is not envy-free. But how do you formalize such a strategy ... it seems a bit odd to me....

I think you're right, it's a problem if players can move their knives in a discontinuous way. I wasn't considering ties for who yells cut, which is fine with continuous knife movements because if you yell at the moment that you are indifferent, then you're ok -- but with discontinuous moves there might not be this moment.

You might also mention more explicitly the other part of your math.se question, that you are not interested in the greedy max-cut approach (or I guess that's what your second paragrap means).

Adding to the question, is random sampling considered better or worse than counting points on a discrete grid?

@Marco, but if you have been moving for a long time, the zombies have been moving for a long time as well and with a coordinated group motion. How do we know that their motion can produce any local configuration after any length of time? (Especially if we do not know the walker's strategy.)