  In this article, we combine sums of squares (SOS) and sums of nonnegative
circuit (SONC) forms, two independent nonnegativity certificates for real
homogeneous polynomials. We consider the convex cone SOS+SONC of forms that
decompose into a sum of an SOS and a SONC form and study it from a geometric
point of view. We show that the SOS+SONC cone is proper and neither closed
under multiplications nor under linear transformation of variables. Moreover,
we present an alternative proof of an analog of Hilbert's 1888 Theorem for the
SOS+SONC cone and prove that in the non-Hilbert cases it provides a proper
superset of both the SOS and the SONC cone. This follows by exploiting a new
necessary condition for membership in the SONC cone.
