  For any flag simplicial complex $\Theta$ obtained by stellar subdividing the
boundary of the cross polytope in edges, we define a flag simplicial complex
$\Gamma(\Theta)$ (dependent on the sequence of subdivisions) whose $f$-vector
is the $\gamma$-vector of $\Theta$. This proves that the $\gamma$-vector of any
such simplicial complex satisfies the Frankl-F\"{u}redi-Kalai inequalities,
partially solving a conjecture by Nevo and Petersen \cite{np}. We show that
when $\Theta$ is the dual simplicial complex to a nestohedron, and the sequence
of subdivisions corresponds to a flag ordering as defined in \cite{ai}, that
$\Gamma(\Theta)$ is equal to the flag simplical complex defined there.
