Convex Forms That Are Not Sums of Squares

  title={Convex Forms That Are Not Sums of Squares},
  author={Grigoriy Blekherman}
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… CONTINUE READING