This submission provides three examples of at least paradoxical phenomena that happen in higher dimensions:
Example A proves that the greatest volume part of a hypercube is concentrated at its corners.
Example B proves that virtually all of the content of a hypersphere is concentrated close to its surface.
Finally, Example C also proves that the probability mass of a multivariate Normal distribution exhibits a rapid
migration into the extreme tails. In very high dimensions, virtually the entire sample will be in the distribution tails!
The theory of these examples was reproduced from the book:
"Multivariate Density Estimation - Theory, Practice, and Visualization" by David W. Scott, 1992, John Wiley & Sons, Inc.
I confirmed the theoretical formulas by use of Monte Carlo simulations because originally I had trouble to believe them!
Ilias Konsoulas (2023). Paradoxical Behavior of Multidimensional Data (https://www.mathworks.com/matlabcentral/fileexchange/53260-paradoxical-behavior-of-multidimensional-data), MATLAB Central File Exchange. Retrieved .
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I have corrected a couple of bugs in hypernormal.m.
Added a nice promo picture.