It is defined as a set of points in real affine (or Euclidean) space of any dimension n, that has flat sides.
This definition of polyhedra is particularly important as it provides a geometric perspective for problems in linear programming.
Other examples include: A subset F of a polyhedron P is called a face of P if there is a halfspace H (defined by some inequality a1Tx ≤ b1) such that H contains P and F is the intersection of H and P.[3]: 9 Suppose P is a polyhedron defined by Ax ≤ b, where A has full column rank.
Then, v is a vertex of P if and only if v is a basic feasible solution of the linear system Ax ≤ b.
The set cone(E) is also called the recession cone of P.[3]: 10 Carathéodory's theorem states that, if P is a d-dimensional polytope, then every point in P can be written as a convex combination of at most d+1 affinely-independent vertices of P. The theorem can be generalized: if P is any d-dimensional polyhedron, then every point in P can be written as a convex combination of points v1,...,vs, v1+e1,...,v1+et with s+t ≤ d+1, such that v1,...,vs are elements of minimal nonempty faces of P and e1,...,et are elements of the minimal nonzero faces of the recession cone of P.[3]: 10 When solving algorithmic problems on polyhedra, it is important to know whether a certain polyhedron can be represented by an encoding with a small number of bits.