Smooth exhaustion functions in convex domains

We show that in every bounded convex domain in $\mathbb R^n$ there exists a smooth convex exhaustion function $\psi$ such that the product of all eigenvalues of the matrix $(\partial ^2\psi /\partial x_j\partial x_k)$ is $\ge 1$. Moreover, if the domain is strictly convex, then $\psi$ can be chosen so that every eigenvalue is $\ge 1$.