Partial group algebra

In mathematics — specifically, in probability theory — the Laplace functional of a metric probability space is an extended-real-valued 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 sets of (X, d). The Laplace functional of (X, d, μ) is the function

$E_{(X,d,\mu )}\colon [0,+\infty )\to [0,+\infty ]$

defined by

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

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

$\alpha _{(X,d,\mu )}(r):=\sup\{1-\mu (A_{r})\mid A\subseteq X{\text{ and }}\mu (A)\geq {\tfrac {1}{2}}\},$

where

$A_{r}:=\{x\in X\mid d(x,A)\leq r\}.$

In this notation,

$\alpha _{(X,d,\mu )}(r)\leq \inf _{\lambda \geq 0}e^{-\lambda r/2}E_{(X,d,\mu )}(\lambda ).$

References

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