Short summary and conventions
Both cones and polyhedra can be described either by giving generators, the socalled Vrepresentation or by giving inequalities, the socalled Hrepresentation. We have the following conventions:
1. Rays, vertices, and generators of the lineality space are given as columns of matrices.
2. Inequalities and hyperplanes are given as rows of matrices.
3. The inequality description of a cone is $A\cdot x\ge 0$.
4. The inequality description of a polyhedron is $A\cdot x\le b$.
Conventions for cones
For cones we have the convention that the scalar product of generators with inequalities is positive:




The hyperplanes of a cone evaluate to zero with the rays of a cone, just like the linealitySpace evaluates to zero with the facets.


Conventions for polyhedra
For a polyhedron the situation is slightly different, as we have a right hand side to take into account, since we are dealing with affine hyperplanes instead of just hyperplanes.




The convention is that for any point $p$ in the polyhedron we have $A\cdot p\le b$. This means we have $0\le b  A\cdot p$. Again, this may be handled differently elsewhere.

From the above convention it follows that the facets evaluate negatively with the rays and linealitySpace of a polyhedron. Conversely to hyperplanes evaluate to constants on the vertices of a polyhedron.









Full representations
1. The pair (rays, linealitySpace) is a valid Vrepresentation of a cone.
2. The pair (facets, hyperplanes) is a valid Hrepresentation of a cone.
3. The triple (vertices, rays, linealitySpace) is a valid Vrepresentation of a polyhedron.
4. The triple (facets, hyperplanes) is a valid Hrepresentation of a polyhedron.
That means we have the following identities:








