What is a dual of a convex polytope?

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

for all

where is assumed to contain the origin in its interior.
When contains the origin in its interior,
the polytope is called the *polar* of . One can
easily show that

for all

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