Concentration of measure
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In mathematics, concentration of measure is a principle that is applied in measure theory, probability and combinatorics, and has consequences for other fields such as Banach space theory. It is illustrated by the phenomenon of the measure near an equatorial section of the n-sphere, where n is large. If one estimates what proportion of the measure on the n-sphere, for the usual rotationally-invariant measure, is near any one (n − 1)-sphere that is equatorial in it (analogous to a great circle on the Earth's surface, when n = 2), the answer is that almost all of the measure is concentrated near it. In other words, the 'poles' and their neighbourhoods down to very small latitudes account for a tiny proportion of the generalised 'surface area', in a sense that can be made precise as n → ∞.
This principle can be made the basis of many proofs; recognition of it is often attributed to Paul Lévy. It was applied by Vitali Milman to prove Dvoretzky's theorem on subspaces of Banach spaces that are almost Hilbert spaces.
[edit] See also
- Gaussian isoperimetric inequality
- Spherical isoperimetric inequality
[edit] Further reading
- Ledoux, Michel (2001). The Concentration of Measure Phenomenon. American Mathematical Society. ISBN 0821828649.
- Devdatt P. Dubhashi and Alessandro Panconesi, Concentration of Measure for the Analysis of Randomised Algorithms (pdf), October 21, 2005 (207 pages)
