Convex sets with semidefinite representation

Mathematical Programming, Series A. Published electronically on May 7th 2008International audienceWe provide a sufficient condition on a class of compact basic semialgebraic sets K 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 in sandwich between co(K) and 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 polynomial f, is a sum of squares. We also provide an approximate SDr specific to the convex case