Concentration of measure

Summary

Concentration of measure is a phenomenon of many high-dimensional metric spaces with a probability measure. On such a space, almost all of the space lies close to every set of measure at least $1/2$. Equivalently, every Lipschitz function is nearly constant, close to its median, outside a set of very small measure. The sphere $S^{n-1}$ and the Gaussian measure 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.