Primal-dual methods for vertex and facet enumeration

Every convex polytope can be represented as the intersection of a finite set of halfspaces as the convex hull of its vertices. Transforming from the halfspace (respectively vertex) to the vertex (respectively halfspace) representation is called vertex enumeration (respectively facet enumeration). An open question is whether there is an algorithm for these two problems (equivalent by geometric duality) that is polynomial in the input size and …