For a Borel subset $B$ of a metric space $X = (X,d)$ and $\epsilon>0$, recall the defintion of the $\epsilon$-blowup of $B$, namely $B^\epsilon = \{x \in X | d(x,B) \le \epsilon\}$. Let $\mu$ be a probability measure on $X$, and suppose $0 < \mu(B) < 1$. Under certain conditions (on $\mu$), we know that $1-\mu(B^\epsilon)$ decreases exponentially with increasing $\epsilon \ge \epsilon_0$, where $\epsilon_0>0$ is some phase-transition point. This is the phenomenon of measure-concentration in metric spaces. For example, Corollary 1.1 of Otto-Villani (2000) shows such a result with $\epsilon_0 = \sqrt{2\log(1/\mu(B))}$ for measures satisfying the *Talagrand transportation-cost inequality*.

# Question

Can one have an unconditional upper bound (maybe something not exponential, but polynomial in $\epsilon$) for $1-\mu(B^\epsilon)$ valid for "small" $\epsilon$, say for all $\epsilon \le \epsilon_0' < \epsilon_0$ ?

# Clarifications

**Unconditional**, meaning not assuming any magical properties on $\mu$, like log-concavity, etc.

**Polynomial**, meaning something like $1-\mu(B^\epsilon) \le \epsilon poly(\epsilon)$.

Unconditional, meaning not assuming any magical properties on $\mu$, like log-concavity, etc.Polynomial, meaning something like $1-\mu(B^\epsilon) \le poly(\epsilon)$. $\endgroup$ – dohmatob Feb 1 at 12:58