Convex sets with semidefinite representation

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