A multivariate polynomial is SOS-convex (or sum of squares convex) if its Hessian matrix H can be factored as H(x) = ST(x)S(x) where S is a matrix (possibly rectangular) which entries are polynomials in x. In other words, the Hessian matrix is a SOS matrix polynomial. An equivalent definition is that the form defined as g(x,y) = yTH(x)y is a sum of squares of forms.
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