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  "body": "\n## Summary\n\nThe curse of dimensionality is the exponential growth in the dimension $d$ of the cost of solving a numerical problem to a fixed accuracy.\nThe phrase is due to Richard Bellman.\nIn integration and approximation it means that the number of function values needed for error $\\varepsilon$ grows like $c^d$ with $c \u003e 1$.\nMethods such as [[sparse grids|84BR7X6HET]], Monte Carlo and quasi-Monte Carlo methods, and restrictions to special classes of functions are ways to avoid it.\n\n## Explanations\n\n### L1 human\n\nTo sample a line segment with spacing $0.1$ one needs $10$ points, a square needs $100$, and a cube needs $1000$.\nIn $d$ dimensions the same spacing needs $10^d$ points, which is hopeless for $d = 50$.\nHigh-dimensional space is so large that any reasonable set of sample points leaves most of it far from every point.\nSo an unknown function can hide almost anything between the samples.\nTake the functions on $[0, 1]^d$ bounded by $1$ and with Lipschitz constant $1$.\nFor example, no method that uses at most $2^{d-1}$ of their values can always compute the integral with error below $1/8$.\n",
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