From Osher Doctorow
Sourav Chatterjee's "Concentration inequalities with exchangeable
pairs, a Dissertation submitted to the department of satistics and the
committee on graduate studies of Stanford University in partial
fulfillment of the requirements for the Degree of Doctor of
Philosophy," math.PR/0507526 v1 26 Jul 2005, tells us (p. 21), tells us
that the idea of the "concentration property" was pioneered by Paul
Levy who observed that for the uniform measure on high dimensional
spheres, sets with measure > = 1/2 "engulf" most of the space when
expanded slightly. It holds in high dimensional spaces for certain
probability measures including products of one-dimensional measures
that are "well-behaved". Amir and Milman and Gromov and Milman in the
1980s (referenced in Chatterjee) revived interest in this and in the
connection of Levy's work with concentration inequalities, which latter
have been a topic of interest in geometric functional analysis and in
convex geometry for the last 30 years. Readers may recall my
occasional references to Benyamini (of Israel) and Lindendrauss' work
on geometric nonlinear functional analysis.
In Chatterjee's notation, the concentration function a_(X,d,u) mapping
[0, infinity) to [0, 1] on a given Polish space (X, d) with a given
probability measure u on X is defined as:
1) a_(X,d,u)(t) = sup{1 - u(A_t): A C X, u(A) > = 1/2}
where I use C for "is a subset of" and A_t is defined as:
2) A_t \ {x e X: d(x, A) < t} (e is "is an element of")
and d(x, A0 is inf{d(x,y): y e A}.
Osher Doctorow
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