What is a dual of a convex polytope?

For a convex polytope $P$, any convex polytope $P'$ with $FL(P')$ anti-isomorphic to $FL(P)$ (i.e. ``upside-down'' of $FL(P)$) is called a (combinatorial) dual of $P$. By the definition, a dual polytope has the same dimension as $P$. The duality theorem states that every convex polytope admits a dual.

Theorem 1 (Duality of Polytopes)   Every nonempty $d$-polytope $P$ in $R^d$ admits a dual polytope in $R^d$. In particular, one can construct a dual polytope by the following ``polar'' construction:

$$P^* = \{y \in R^d: x^T y \le 1$$    for all $$x \in P \}$$

where $P$ is assumed to contain the origin in its interior.

When $P$ contains the origin in its interior, the polytope $P^*$ is called the polar of $P$. One can easily show that

$$P^* = \{y \in R^d: v^T y \le 1$$    for all $$v \in V(P) \}$$

where $V(P)$ denote the set of vertices of $V$, and this inequality (H-) representation of $P^*$ is minimal (i.e. contains no redundant inequalities).