When sampling from a high dimensional Gaussian distribution the results are similar to sampling from a uniform distribution on a unit sphere. Upon searching, I was able to find this mentioned in [1].
This phenomenon already begins in 3D, see [2].
My objective is to construct a distribution in high dimensions such that the property of Gaussians in 1D and 2D holds, mainly that when given a random sample from such a desired distribution, the best guess as to the mean of that distribution is that given sample. And so that there would be a "falloff" from the mean.
How can this be achieved? Can it be done?
[1] https://www.inference.vc/high-dimensional-gaussian-distributions-are-soap-bubble/
[2] https://en.wikipedia.org/wiki/Proof_of_Stein%27s_example
Edit: After some more keyword googling, I was able to find this very recent Twitter thread which further expands on the problem: https://twitter.com/johncarlosbaez/status/1298274201682325509