Certificates of convexity for basic semi-algebraic sets

We provide two certificates of convexity for arbitrary basic semi-algebraic sets of ${\mathbb{R}}^n$. The first one is based on a necessary and sufficient condition whereas the second one is based on a sufficient (but simpler) condition only. Both certificates are obtained from any feasible solution of a related semidefinite program and so can be obtained numerically (however, up to machine precision).