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We show that for any metric probability space $(M,d,\mu)$ with a subgaussianconstant $\sigma^2(\mu)$ and any set $A \subset M$ we have $\sigma^2(\mu_A)\leq c \log\left(e/\mu(A)\right)\,\sigma^2(\mu)$, where $\mu_A$ is arestriction of $\mu$ to the set $A$ and $c$ is a universal constant. As aconsequence we deduce concentration inequalities for non-Lipschitz functions.

Concentration Properties of Restricted Measures with Applications to Non-Lipschitz Functions