N-dimensional polyhedron

An n-dimensional polyhedron is a geometric object that generalizes the 3-dimensional polyhedron to an n-dimensional space. It is defined as a set of points in real affine (or Euclidean) space of any dimension n, that has flat sides. It may alternatively be defined as the intersection of finitely many half-spaces. Unlike a 3-dimensional polyhedron, it may be bounded or unbounded. In this terminology, a bounded polyhedron is called a polytope.

Analytically, a convex polyhedron is expressed as the solution set for a system of linear inequalities, a_i^Tx ≤ b_i, where a_i are vectors in Rⁿ and b_i are scalars. This definition of polyhedra is particularly important as it provides a geometric perspective for problems in linear programming.^: 9