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- TL;DR Summary
- The volume outside a hypersphere inscribed in a unit hypercube converges to 1 as the dimension increases. It seems that the space somehow “gets bigger” as the dimension increases. However, ##\mathbb{R}## has the same cardinality as ##\mathbb{R^2}##.
The volume outside a hypersphere inscribed in a unit hypercube converges to 1 as the dimension increases. It seems that the space somehow “gets bigger” as the dimension increases. However, given data points in the interval ##[0,1]##, the density of points in the interval compared to the density of points for the same points in the unit square doesn’t increase, since ##\mathbb{R}## has the same cardinality as ##\mathbb{R^2}##. Then, what is it that “gets bigger”, gets more? Is sparseness a more relevant term than density?