Suppose a polytope $P$ is contained in its dual polytope $\tilde{P}$. Does there always exist a polytope $Q$ that contains $P$ and is self dual $Q=\tilde{Q}$? Is there any bound on the minimal number of vertices that $Q$ should have given the number of vertices of $P$? Is there any algorithm for finding such a polytope?

The dual of a subset $A$ of a vector space $V$ is defined as $\tilde{A}:=\{\vec{y}\in V |\vec{y}\cdot\vec{x}\geq -c\; \forall \vec{x}\in A\}$, where $c$ is a given positive constant, and $\vec{y}\cdot\vec{x}$ is the scaler product between $\vec{y}$ and $\vec{x}$.