Polyhedral faces in Gram spectrahedra of binary forms

We analyze both the facial structure of Gram spectrahedron $\mathrm{Gram}(f)$ and Hermitian $\mathcal{H}^{\scriptscriptstyle+}(f)$ a nonnegative binary form $f \in \mathbb{R}[x, y]_{2d}$. show that if $F \subseteq \mathcal{H}^{\scriptscriptstyle+}(f)$ is polyhedral face dimension $k$ then $\binom{k+1}{2} \leq d$. Conversely, for all $k \mathbb{N}$ $d \geq \binom{k+1}{2}$ we general positive y]_{2d}$ with distinct roots contains $F$ which $k$-simplex whose extreme points are rank-one tensors. For (k+1)^2$ (symmetric) $(\mathrm{rk}(F), \dim(F)) = (2(k+1), k)$.