The conditions you need for that to be true are substantially weaker than being Euclidean (though, when people are talking about "weird" behavior in high-dimens

The conditions you need for that to be true are substantially weaker than being Euclidean (though, when people are talking about "weird" behavior in high-dimensional spaces nowadays, it's in the context of ML and Euclidean stuff anyway). If you have a meaningful notion of dimension (basic properties like <1,0> being different from <0,1> and <1,0> being closer to <1,1> than <0,1>) and and don't have discretized shenanigans (which would collapse the inequality I'm exploiting to a strict equality, with some sort of 1^n=1 behavior) then the natural measure induced by the metric in question will exhibit the described behavior. You can easily verify that for every metric represented in scipy or whatnot, and a proper proof isn't too much more work.