A complete characterization of the gap between convexity and sos-convexity

Our first contribution in this paper is to prove that three natural sum of squares (sos) based sufficient conditions for convexity of polynomials, via the definition of convexity, its first order characterization, and its second order characterization, are equivalent. These three equivalent algebraic conditions, henceforth referred to as sos-convexity, can be checked by semidefinite programming, whereas deciding convexity is NP-hard. If we denote the set of convex and sos-convex polynomials in n variables of degree d with C̃_n,d and ∑C̃_n,d respectively, then our main contribution is to prove that C̃_n,d = ∑C̃ _n,d if and only if n = 1 or d = 2 or (n, d) = (2, 4). We also present a complete characterization for forms (homogeneous polynomials) except for the case (n, d) = (3, 4), which is joint work with Blekherman and is to be published elsewhere. Our result states that the set C_n,d of convex forms in n variables of degree d equals the set ∑C_n,d of sos-convex forms if and only if n = 2 or d = 2 or (n, d) = (3, 4). To prove these results, we present in particular explicit examples of polynomials in C̃_2,6\ ∑C̃_2,6 and C̃_3,4\∑C̃_3,4 and forms in C_3,6\∑C_3,6 and C_4,4\ ∑C_4,4, and a general procedure for constructing forms in C _n,d+2\∑C_n,d+2 from nonnegative but not sos forms in n variables and degree d. Although for disparate reasons, the remarkable outcome is that convex polynomials (resp., forms) are sos-convex exactly in cases where nonnegative polynomials (resp., forms) are sums of squares, as characterized by Hilbert.