The ring of combinatorial convex polytopes developed in the works of Victor M. Buchstaber and the author gave new view on the classical problem of characterization of all the integral vectors that are vectors of flag numbers of convex polytopes. Many results and constructions in this area such as the Bayer-Billera equations, the toric $g$-polynomial, the $cd$-index obtained new interpretation that led to new results.

Our aim is to build new operators on the ring of convex polytopes that have a nice geometric interpretation. The main result is the following.

Let $P$ be a convex polytope in $\matbbb R^n$ and $l$ be a line through the origin in general position with respect to $P$. For any facet $F$ of $P$ consider the intersection $x_F$ of $l$ with the supporting hyperplane of $F$. For the top facet $F_t$ and for the bottom facet $F_b$ we have  $x_F\in F$ and for all the others  $x_F\notin F$. In this case consider the projection $\pi_F(F)$ of the facet $F$ from the point $x_F$ to the hyperplane through the origin orthogonal to $l$. Take the sum $K(P)=\sum_{F\ne F_t, F_b} \pi_F(F)$ in the ring of polytopes.

The $cd$-index is a non-commutative polynomial in $c$ and $d$, $\deg c=1$, $\deg d=2$, invented by Jonathan Fine. It captures all the information on the flag numbers of a polytope.

{\bf Theorem.} {\it For the $cd$-index $\Xi$ we have $\Xi(P)=Ac+K(P)d$ for some non-commutative polynomial A. Moreover, $2A=Xi((d-CK)P)$, where $dP$ is the sum of all the facets of $P$ and $CP=2{\rm pyr}(P)-{\rm bipyr}(P)$. }

The first part of this result is obtained via polarity from the geometric interpretation of the cd-index by Carl Lee. The second part is obtained by the relations on the $cd$-index and the $C=2{\rm pyr}-{\rm bipyr}$ and $D={\rm bipyr}{\rm  pyr}-{\rm pyr}{\rm  bipyr}$ operations on the ring of polytopes.