  Carbery proved that if $u:\mathbb{R}^n \rightarrow \mathbb{R}$ is a positive,
strictly convex function satisfying $\det D^2u \geq 1$, then we have the
estimate $$ \left| \left\{x \in \mathbb{R}^n: u(x) \leq s \right\} \right|
\lesssim_n s^{n/2}$$ and this is optimal. We give a short proof that also
implies other results. Our main result is an estimate for the sublevel set of
functions $u:[0,1]^2 \rightarrow \mathbb{R}$ satisfying $1 \leq \Delta u \leq
c$ for some universal constant $c$: for any $\alpha > 0$, we have $$ \left|
\left\{x \in [0,1]^2 : |u(x)| \leq \varepsilon\right\}\right| \lesssim_{c}
\sqrt{\varepsilon} + \varepsilon^{\alpha - \frac12}
\int_{[0,1]^2}{\frac{|\nabla u|}{|u|^{\alpha}} dx}.$$ For 'typical' functions,
we expect the integral to be finite for $\alpha < 1$. While
Carbery-Christ-Wright have shown that no sublevel set estimates independent of
$u$ exist, this result shows that for 'typical' functions satisfying $\Delta u
\geq 1$, we expect the sublevel set to be $\lesssim \varepsilon^{1/2-}$. We do
not know whether this is sharp or whether similar statements are true in higher
dimensions.
