---
tag: CFZ1J559MK
kind: concept
name: Concentration of measure
msc: [52A23, "60E15", 60B11]
sources:
  - url: https://en.wikipedia.org/wiki/Concentration_of_measure
same_as:
  wikidata: Q5158327
  wikipedia: Concentration of measure
---

## Summary

Concentration of measure is a phenomenon of many high-dimensional [[metric spaces|0VKD1TP7F3]] with a [[probability measure|059TJN0NZ1]].
On such a space, almost all of the space lies close to every set of measure at least $1/2$.
Equivalently, every [[Lipschitz|0AW1E6YTJY]] function is nearly constant, close to its median, outside a set of very small measure.
The sphere $S^{n-1}$ and the [[Gaussian measure|P0DV2TCH1N]] on $\R^n$ are the model examples.
Milman put the phenomenon forward in the local theory of Banach spaces.
It is now a basic tool in asymptotic convex geometry, probability, statistics and computer science.

## Explanations

### L1 human

On a sphere in very high dimension, almost all of the area lies near the equator, whichever equator one picks.
So if a set covers half the sphere, then a thin band around it already covers almost everything.
In the same way, a function that does not change too fast takes nearly the same value at almost every point.
For example, the average of many independent coin flips is very close to $1/2$ with overwhelming probability, because it depends only a little on each flip.