I need some help going between two equivalent definitions.
First some notation :
$\bullet$ For $A\subseteq \mathcal{X}$ and $r>0$ define what is called the r-blowup of $A$ as \begin{equation} A_r = \{ x \in \mathcal{X} : d(x,A)<r \}. \end{equation}
$\bullet$ We have a metric probability space $(\mathcal{X},d,\mu)$, i.e $(\mathcal{X},d)$ is a Polish space, and $\mu$ a probability measure on its Borel sets.
Now I am going to give two definitions for $\mu$ satisfying a Gaussian concentration of measure. Can anyone show how to go from one definition to the other (of course with a change of constants).
$\textbf{Def.1}$ $\mu$ is said to have Guassian concentration on $(\mathcal{X},d)$ if there exists $K,\kappa >0 $ such that
\begin{equation}\label{Gauss concentration} \textit{whenever}~~\mu(A)\geq \frac{1}{2} ~~ \textit{it implies} ~~ \mu(A_r)\geq 1-K e^{-\kappa r^2} \end{equation}
$\textbf{Def.2}$ We say $\mu$ has Gaussian concentration of measure if there exists $K,\kappa >0$ and some $r_0>0$, such that \begin{equation} \textit{whenever}~~\mu(A)\geq \frac{1}{2} ~~ \textit{it implies} ~~ \mu(A_r) \geq 1-Ke^{-\kappa(r-r_0)^2} ~~ \forall r\geq r_0 \end{equation}
[1, Page 103] claims that we can go between the two with a change of constants, how? Note going from $\textbf{Def.1}$ to $\textbf{Def.2}$ is fine just replace $r$ by $r-r_0$.
[1] Raginsky, Sason. Concentration of Measure Inequalities in Information Theory, Communications and Coding. 2014
As you mentioned, it is easy to go from Definition $1$ to Definition $2$.
Let $K$, $\kappa$, and $r_0$ be the constants in Definition 2. We will now find constants $K'$ and $\kappa'$ for Definition $1$. Set $K' := \max(K,1)e^{4 \kappa' r_0^2}$ and $\kappa':= \kappa/4$ to be the constants in Definition 1. We will show that these constants suffice by considering two cases:
Case 1: $r \in [0,2r_0]$. We have that $$ K' \exp(- \kappa'r^2) = \max(K,1)\exp(\kappa'(4r^2_0 - r^2)) \geq 1, $$ and thus the desired inequality $\mu(A_r)\geq 1 - K' \exp(- r^2)$ holds trivially.
Case 2: $r \geq 2r_0$. We have the following series of inequalities: $$ \begin{aligned}1 - \mu(A_r) &\leq K e^{- \kappa (r - r_0)^2} \\ &\leq K e^{- \kappa r^2/4}\\ &\leq K'e^{- \kappa r^2/4} = K'e^{- \kappa' r^2}, \end{aligned} $$ where we used that $K \leq K'$ and $|r - r_0| \geq r/2 $ for $r \geq 2r_0$.