convex polytope as the convex hull of a finite set of points, where the finite set must contain the set of extreme points of the polytope. Such a definition is called a vertex representation (V-representation or V-description). For a compact convex polytope, the minimal V-description is unique and it is given by the set of the vertices of the polytope. A convex polytope is called an integral polytope if all of its vertices have integer coordinates. Intersection of half-spaces A convex polytope may be defined as an intersection of a finite number of half-spaces. Such definition is called a half-space representation (H-representation or H-description). There exist infinitely many H-descriptions of a convex polytope. However, for a full-dimensional convex polytope, the minimal H-description is in fact unique and is given by the set of the facet-defining halfspaces. A closed half-space can be written as a linear inequality: where is the dimension of the space containing the polytope under consideration. Hence, a closed convex polytope may be regarded as the set of solutions to the system of linear inequalities: where is the number of half-spaces defining the polytope. This can be concisely written as the matrix inequality: where is an matrix, is an column vector whose coordinates are the variables to , and is an column vector whose coordinates are the right-hand sides to of the scalar inequalities. An open convex polytope is defined in the same way, with strict inequalities used in the formulas instead of the non-strict ones. The coefficients of each row of and correspond with the coefficients of the linear inequality defining the respective half-space. Hence, each row in the matrix corresponds with a supporting hyperplane of the polytope, a hyperplane bounding a half-space that contains the polytope. If a supporting hyperplane also intersects the polytope, it is called a bounding hyperplane (since it is a supporting hyperplane, it can only intersect the polytope at the polytope's boundary). The foregoing definition assumes that the polytope is full-dimensional. In this case, there is a unique minimal set of defining inequalities (up to multiplication by a positive number). Inequalities belonging to this unique minimal system are called essential. The set of points of a polytope which satisfy an essential inequality with equality is called a facet. If the polytope is not full-dimensional, then the solutions of lie in a proper affine subspace of and the polytope can be studied as