Convex Forms That Are Not Sums of Squares
نویسنده
چکیده
An orbitope is the convex hull of an orbit of a point under the action of a compact group. We derive bounds on volumes of sections of polar bodies of orbitopes, extending methods developed in [BB03]. As an application we realize the cone of convex forms as a section of the cone of nonnegative bi-homogeneous forms and estimate its volume. A convex form has to be nonnegative, but it has not been previously shown that there exist convex forms that are not sums of squares. Combining with the bounds of [Bl06] we show that if the degree is fixed then the cone of convex forms has asymptotically same size as the cone of nonnegative forms and it is significantly larger asymptotically than the cone of sums of squares. This implies existence of convex forms that are not sums of squares, although there are still no known examples.
منابع مشابه
Matrix Completion , Free Resolutions , and Sums of Squares
Goal: Describe the image of the cone Sn ≥� of positive semidefinite quadratic forms under the projection πG . Sn ≥�: convex cone of quadratic forms∑i , j ai jxix j such that∑i , j ai jpi p j ≥ � for all (p�, . . . , pn) ∈ Rn. Theorem (Diagonalization ofQuadratic Forms). Aquadratic form q ∈ R[x�, . . . , xn] is positive semidefinite if and only if it is a sum of squares of linear forms after a c...
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