Polyhedron

A geometric set formed by intersecting finitely many half-spaces and hyperplanes. It may be bounded or unbounded, and need not have full dimension.

Half-space: One of the two regions into which a hyperplane divides a space, including the hyperplane boundary. A finite intersection of half-spaces and hyperplanes defines a polyhedron.
H-representation: A description of a polyhedron as the solution set of finitely many linear inequalities and equations. It directly encodes the half-spaces and hyperplanes in the definition.
Linear programming: Optimization of a linear objective over variables constrained by linear inequalities and equalities. Its feasible regions are polyhedra, and bounded optima occur at extreme points.
Polytope: A bounded polyhedron; equivalently, the convex hull of finitely many points. Boundedness distinguishes polytopes from general polyhedra.
Hyperplane: A flat affine subspace of codimension one, such as a line in a plane or a plane in three-dimensional space. Hyperplanes supply both boundary constraints and equality constraints in a polyhedron's definition.
V-representation: A description of a polyhedron using convex combinations of points and nonnegative combinations of directions. It describes the same set through its vertices and unbounded directions.
Integer programming: Optimization over linear constraints when some or all decision variables must be integers. Polyhedral relaxations bound and help solve problems with discrete variables.
Cone: A set closed under multiplication by nonnegative scalars and, for a convex cone, under addition. A polyhedral cone is a special unbounded polyhedron with its apex at the origin.
Convex set: A set containing the entire line segment between any two of its points. Intersections of half-spaces are convex, so every polyhedron has this property.
Extreme point: A point in a convex set that cannot be expressed as a nontrivial convex combination of two distinct points in the set. Extreme points are the vertices singled out by a polyhedron's geometry.

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