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The author takes care to make the assumption that A and Z are _global_ minima and M is a _global_ maximum. But, in reading the proof, it seems not necessary to
by Chinjut 10y ago
The author takes care to make the assumption that A and Z are _global_ minima and M is a _global_ maximum. But, in reading the proof, it seems not necessary to make such strong assumptions; would it not work just as well for A and Z to be merely local minima and M to be a local maximum? (For that matter, it should work just as well for A, Z, and M to all merely be local extrema of either sort, so long as A and Z match)
- skj 10y agoIf A or Z aren't global minima, there might be a spot between A and M that is lower than all those between Z and M, and therefore no path.
- Chinjut 10y agoYes, I realize now my error: I was reasoning in the degree calculations for the relevant graph as if "local minimum" was synonymous with "valley", and similarly for "local maximum" and "peak", but of course A and Z are examples of local minima that are not typical valleys (having escape routes on only 1 instead of 2 sides), and a similar point applies to M (construed as impassible). In fairness, the article itself is glib about this important point (it does not discuss the degree calculation for (A, X), (X, Z), (M, X), and (X, M) in general, only for (A, Z) and (M, M) in particular).