  The concentration of measure phenomenon in Gauss' space states that every
$L$-Lipschitz map $f$ on $\mathbb R^n$ satisfies \[ \gamma_{n} \left(\{ x : |
f(x) - M_{f} | \geqslant t \} \right) \leqslant 2 e^{
  - \frac{t^2}{ 2L^2} }, \quad t>0, \] where $\gamma_{n} $ is the standard
Gaussian measure on $\mathbb R^{n}$ and $M_{f}$ is a median of $f$. In this
work, we provide necessary and sufficient conditions for when this inequality
can be reversed, up to universal constants, in the case when $f$ is
additionally assumed to be convex. In particular, we show that if the variance
${\rm Var}(f)$ (with respect to $\gamma_{n}$) satisfies $ \alpha L \leqslant
\sqrt{ {\rm Var}(f) } $ for some $ 0<\alpha \leqslant 1$, then \[ \gamma_{n}
\left(\{ x : | f(x) - M_{f} | \geqslant t \}\right) \geqslant c e^{
  -C \frac{t^2}{ L^2} } , \quad t>0 ,\] where $c,C>0$ are constants depending
only on $\alpha$.
