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We consider a wide class of closed convex cones $K$ in the space of real$n\times n$ symmetric matrices and establish the existence of a chain of facesof $K$, the length of which is maximized at $\frac{n(n+1)}{2} + 1$. Examples ofsuch cones include, but are not limited to, the completely positive and thecopositive cones. Using this chain, we prove that the distance to polyhedralityof any closed convex cone $K$ that is sandwiched between the completelypositive cone and the doubly nonnegative cone of order $n \ge 2$, as well asits dual, is at least $\frac{n(n+1)}{2} - 2$, which is also the worst-casescenario.

On the longest chain of faces of completely positive and copositive cones