[CuriouslyC]

You don't need a complicated proof, just assume a distribution in some very high number of dimensions, with samples from that distribution having randomly generated values from the distribution for each dimension. If you have if you have ~300 dimensions then statistically at least one dimension will be ~3SD from the mean, i.e. "on the edge," and as long as any one dimension is close to an edge, we define a point as being "near the edge."

It's not really meaningful though, at high dimensions you want to consider centrality metrics.