Suppose that $v$ is a convex function on ${\bf R}^{n}$, attaining minimal value $0$, and suppose $\det(\nabla^{2}v)\geq\lambda$ when $v\leq 1$. Then if $K$ is the set where $v\leq 1$ we have ${\rm Vol}(K)\leq C\lambda^{-1/2}$ for some constant $C$ depending only on the dimension $n$.