Smooth exhaustion functions in convex domains

Source abstract

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