Fiber bodies of Gram spectrahedra

Given a compact, convex set $K\subset \mathbb{R}^{n+m}$. Consider the projection of $K$ to the first $n$ coordinates. Over every point in $\mathbb{R}^n$ we have a convex fiber. The fiber body is "the average" over all such fibers. Gram spectrahedra are precisely the fibers of a linear map where $K$ is the cone of positive semidefinite matrices. Therefore, the fiber body of Gram spectrahedra is "the average" Gram spectrahedron in some sense. We study the boundary of fiber bodies for Gram spectrahedra of binary/ternary forms of low degree. (This is joint work in progress with Chiara Meroni.)