OK, Another reference is [1], that agrees with the result given by the OP. I'm not trying to do research here, I'm just boggling at the unintuitive result, and

OK, Another reference is [1], that agrees with the result given by the OP. I'm not trying to do research here, I'm just boggling at the unintuitive result, and trying to see if there might be a flaw in the chain of logic. The fact that this is "well known" is enough to scare me off from barking up this particular tree. [1] https://www.math.wustl.edu/~feres/highdim https://www.math.wustl.edu/~feres/highdim