Convex sets with semidefinite representation

We provide a sufficient condition on a class of compact basic semialgebraic sets K ⊂ R for their convex hull co(K) to have a semidefinite representation (SDr). This SDr is explicitly expressed in terms of the polynomials gj that define K. Examples are provided. We also provide an approximate SDr; that is, for every fixed ! > 0, there is a convex set K! such that co(K) ⊆ K! ⊆ co(K) + !B (where B is the unit ball of R), and K! has an explicit SDr in terms of the gj ’s. For convex and compact basic semi-algebraic sets K defined by concave polynomials, we provide a simpler explicit SDr when the nonnegative Lagrangian Lf associated with K and any linear f ∈ R[X] is a sum of squares. We also provide an approximate SDr specific to the convex case.