I reserve the possibility that I am completely mistaken, in which case I apologize in advance, on the condition that you refer me to an actual calculation or derivation (not just a quote in the same spirit) in a paper or text or textbook.

I have seen such insinuation multiple times, that samples "collect" near the unit d-sphere when samples are drawn from the unit d-sphere in high d dimensions.

From a physics perspective this is very familiar to me, but not in the sense of fact, but in the sense of misinterpretation.

I do believe the observation is very useful in the educational sense as long as it is pointed out as being a paradoxical illusion. In this sense I can appreciate (even encourage) a professor or a TA showing this phenomenon, on the condition they finish up by explaining why this seems to be the case, but is nevertheless a misinterpretation, and they should make sure the connection with Jacobian determinants etc are made clear.

Consider a normal distribution of any dimension (as high as you want), but I will showcase the phenomenon even with low dimensions (here merely d=3) to illustrate this has nothing to do with high dimensions.

Clearly the probability density is maximal in the center of the distribution.

In computer processing of data points, we typically loop over points, calculate some hopefully interesting function on each sample, and then plot the samples say by binning with equal bin sizes. A programmer typically disregards transformation properties like the Jacobian determinant. Suppose the value we calculate for each data point is the absolute length or distance from the center. The further we go from the center the smaller the probability density of the normal distribution becomes... but the larger the volume of a shell of radius r!

Since we are binning with equal bin sizes (equal length intervals per bin), for small lengths, then even though the actual probability density is highest near the center, we will get relatively few samples because the volume under consideration is small compared to the volume under consideration for a shell of a larger radius of equal thickness (area of a sphere grows quadratically with radius). However for even larger distances, the exponential decay of the normal distribution dominates and the number of samples in highest radius bins will decrease again. So in between there will be a peak.

This explains the fact that [ the probability density of [ the absolute distance from the center over [ the sample points ] ] has a peak at some non-zero length.

But it is a conceptual mistake to interpret this as if those sample points in the original d-dimensional space form a dense shell on some "unit sphere" ... This is a pure illusion which illustrates the interpreter is not familiar with jacobian determinants etcetera.

Consider volume or triple integrals over some volume element dx dy dz and for symmetry you prefer integrating in a spherically symmetric coordinate system, theta,phi,r then you can not simply replace dx dy dz with dtheta dphi dr, you need to use dV = dx dy dz = r^2 sin( phi ) dtheta dphi dr.

It is this necessary factor that is ignored when processing sample by sample and causing this illusion in the AI community. I did not follow conventional machine learning courses, but given the learned language used whenever I see statements to the effect of samples lying near the unit sphere in high dimensions, I can only conclude it has its origins in 1) direct observation or experience of plotting in bins of the length of the vector without guidance in interpretation; or 2) guidance during education, where the phenomenon is shown, and the origin of the paradoxical illusion or confusion adequately explained and then the illusory nature subsequently forgotten or 3) a teachers assistant having gone through 2) and showing the phenomenon to students without emphasizing the illusory nature of the misinterpretation.

But perhaps I am wrong, and the normal distribution in high dimensions actually has a higher probability density near its "unit sphere" if the dimension is beyond some critical dimension d_c... but again, I'd like to see a derivation showing it :)

EDIT: For more precise language: they do "collect" (reach a peak) at a certain non-zero length or distance, but they do not collect to a unit sphere of such radius in the original space!