see also - the section titled "Distance function" from https://en.wikipedia.org/wiki/Curse_of_dimensionality https://en.wikipedia.org/wiki/Curse_of_dimensionali

see also - the section titled "Distance function" from https://en.wikipedia.org/wiki/Curse_of_dimensionality https://en.wikipedia.org/wiki/Curse_of_dimensionality > One way to illustrate the "vastness" of high-dimensional Euclidean space is to compare the proportion of an inscribed hypersphere with radius r and dimension d, to that of a hypercube with edges of length 2 r . [...] As the dimension d of the space increases, the hypersphere becomes an insignificant volume relative to that of the hypercube E.g. in two dimensions the ratio area of the circle of radius r=1 to the area of a square of side length 2r is pi / 4 = approx 0.785. As the number of dimensions becomes large, this ratio drops toward zero. There's also some discussion of how using distance functions to measure distances from reference points become less useful as the number of dimensions of the space increases -- all distances becoming similar -- although > research has shown this to only hold in the artificial scenario when the one-dimensional distributions are independent and identically distributed.