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In [[mathematics]] — specifically, in [[probability theory]] — the '''Laplace functional''' of a metric probability space is an [[extended real number line|extended-real-valued]] [[function (mathematics)|function]] that is closely connected to the [[concentration of measure]] properties of the space.

==Definition==

Let (''X'', ''d'', ''μ'') be a metric probability space; that is, let (''X'', ''d'') be a [[metric space]] and let ''μ'' be a [[probability measure]] on the [[Borel set]]s of (''X'', ''d''). The '''Laplace functional''' of (''X'', ''d'', ''μ'') is the function

:<math>E_{(X, d, \mu)} \colon [0, +\infty) \to [0, +\infty]</math>

defined by

:<math>E_{(X, d, \mu)}(\lambda) := \sup \left\{ \left. \int_{X} e^{\lambda f(x)} \, \mathrm{d} \mu(x) \right| f \colon X \to \mathbb{R} \text{ is bounded, 1-Lipschitz and has } \int_{X} f(x) \, \mathrm{d} \mu(x) = 0 \right\}.</math>

==Properties==

The Laplace functional of (''X'', ''d'', ''μ'') can be used to bound the concentration function of (''X'', ''d'', ''μ''). Recall that the '''concentration function''' of (''X'', ''d'', ''μ'') is defined for ''r'' > 0 by

:<math>\alpha_{(X, d, \mu)}(r) := \sup \{ 1 - \mu(A_{r}) \mid A \subseteq X \text{ and } \mu(A) \geq \tfrac{1}{2} \},</math>

where

:<math>A_{r} := \{ x \in X \mid d(x, A) \leq r \}.</math>

In this notation,

:<math>\alpha_{(X, d, \mu)}(r) \leq \inf_{\lambda \geq 0} e^{- \lambda r / 2} E_{(X, d, \mu)}(\lambda).</math>

==References==

* {{cite book
| last = Ledoux
| first = Michel
| title = The Concentration of Measure Phenomenon
| series = Mathematical Surveys and Monographs
| volume = 89
| publisher = American Mathematical Society
| location = Providence, RI
| year = 2001
| pages = x+181
| isbn = 0-8218-2864-9
}} {{MathSciNet|id=1849347}}

[[Category:Probability theory]]
[[Category:Metric geometry]]

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en>Michael Hardy: /* References */ endash