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.
convex sets semidefinite representation sum of squares representation of positive polynomials