A complete characterization of the gap between convexity and sos-convexity (Q2848175)

The authors approach the problem of describing the convexity of polynomials based on sum of squares, proving that three types of sufficient conditions are equivalent: the definition of convexity, its first order characterization and its second order characterization. More, they focus on the famous problem discussed by \textit{D. Hilbert} [Math. Ann. 32, 342--350 (1888; JFM 20.0198.02)] on the existence of nonnegative polynomials that are not expressible as sum of squares of polynomials. The class of polynomials that admit a sum of squares decomposition of the Hessian matrix, known as sos-convex according to \textit{J. W. Helton} and \textit{J. Nie} [Math. Program. 122, No. 1 (A), 21--64 (2010; Zbl 1192.90143)], is discussed in relation with the classic convexity property. One of the main results of the paper is that the set of convex polynomials in \(n\) variables of degree \(d\) coincides with the set of sos-convex polynomials in \(n\) variables of degree \(d\) if and only if \(n=1\) or \(d=2\) or \((n,d)=(2,4)\). More profound results are derived in the class of homogeneous polynomials, in which the coincidence of the two convexities occurs if and only if \(n=2\) or \(d=2\) or \((n,d)=(3,4)\). The comparison between nonnegativity and convexity of polynomials concludes the paper by the following list of similarities: (i) both nonnegativity and convexity are properties that hold only for even degree polynomials; (ii) nonnegativity is equivalent to convexity in case of quadratic forms; (iii) both notions are NP-hard to check exactly for degree \(4\) and larger and (iv) nonnegativity is equivalent to sum of squares exactly in dimensions and degrees where convexity is equivalent to sos-convexity. Few questions and open problems are formulated in this context.