Colin

@Cat Lady Margaret I think it simply lacks an end as opposed to a normal STRIP. Also, a Möbius strip is often metaphorically used for the concept of infinity; in other words, it has no END. One EDGE doesn't have quite the philosophical appeal.

@B @Petrol Yes, I think you have a valid point but what makes the clue fine is that, both idiomatically and in real life, typically skaters venture out onto thin ice while non-skaters do not enter the ice at all, so they do not have to worry about it.

@Andrzej The reasons you listed are also why Tuesday is my least favourite. Monday I can speed-solve, Thursday through Sunday offer a challenge, and Tuesday and Wednesday are left in the "meh" area where there isn't much of a challenge but I can't usually breeze straight through.