Refined Estimates on the Dimensions of Maximal Faces of Completely Positive Cones

The structure of maximal faces of the cone of completely positive matrices is still not well understood in higher dimensions, mainly due to the lack of a general characterization of extreme exposed rays of the copositive cone beyond small matrix orders. This paper contributes to the study of maximal faces of the cone of completely positive matrices by establishing sharper bounds on their dimensions than those currently available. For every odd dimension $n$, we prove that the exact lower bound on the dimensions of maximal faces of the cone of $n\times n$ completely positive matrices equals $n$. For even dimensions $n\ge 8$, we derive a new upper estimate for this lower bound and show that it lies between $n$ and $n+3$. These results substantially refine the previously known bounds.