Laplace functional

Definition for probability measures

For some metric probability space (X, d, μ), where (X, d) is a metric space and μ is a probability measure on the Borel sets of (X, d), the Laplace functional:

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

The Laplace functional maps from the positive real line to the positive (extended) real line, or in mathematical notation:

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

Applications

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

${\displaystyle {\alpha }_{(X,d,\mu )}(r):=sup\{1-\mu (A_r)\mid A\subseteq X\text{ and }\mu (A)\ge \frac{1}{2}\},}$

where

${\displaystyle A_r:=\{x\in X\mid d(x,A)\le r\}.}$

The Laplace functional of (X, d, μ) then gives leads to the upper bound:

${\displaystyle {\alpha }_{(X,d,\mu )}(r)\le \underset{\lambda \ge 0}{inf}{e}^{-\lambda r/2}{E}_{(X,d,\mu )}(\lambda ).}$