Half-space (geometry)

A half-space is defined by a linear inequality. In an n-dimensional real vector space, a closed half-space is the set {x : a·x ≤ b} for some nonzero vector a and scalar b; the open half-space uses a strict inequality. The boundary hyperplane is {x : a·x = b}. Any convex set can be expressed as the intersection of half-spaces, a result known as the supporting hyperplane theorem.¹ In topology, a half-space is homeomorphic to the whole space, but it is not a manifold with boundary unless the boundary is included; the closed half-space is a manifold with boundary, with the boundary being the hyperplane itself.

Half-spaces are the atoms of convex geometry. A convex polytope is the intersection of finitely many closed half-spaces, a representation known as the H-description. Linear programming relies on this: the feasible region is a polyhedron defined by half-space constraints, and the optimal solution occurs at a vertex, which is an extreme point of the intersection.² In computational geometry, the half-space intersection problem asks for the region common to a set of half-spaces; efficient algorithms exist for low dimensions. The concept also underpins the separation theorems, such as the hyperplane separation theorem, which states that two disjoint convex sets can be separated by a hyperplane, each lying in a different half-space.

In functional analysis, a half-space in an infinite-dimensional normed space is defined by a continuous linear functional, and the Hahn–Banach theorem guarantees the existence of supporting half-spaces for convex sets. In order theory, a half-space is a principal ideal or filter in a partially ordered set, generalizing the geometric notion. In discrete geometry, the concept of a half-space is used in the study of arrangements of hyperplanes, where the cells are intersections of half-spaces. The notion also appears in machine learning, where a perceptron is a linear classifier that separates data by a hyperplane, effectively assigning points to one of two half-spaces.³

In number theory, half-spaces appear in the geometry of numbers, where Minkowski's theorem uses symmetric convex bodies, often defined by half-spaces, to prove results on lattice points. In tropical geometry, a tropical half-space is defined by a tropical linear inequality, and its structure differs markedly from the classical case. In the study of Coxeter groups, the reflecting hyperplanes divide space into chambers, each chamber being an intersection of half-spaces; the group acts transitively on the chambers. In robotics, configuration space obstacles are often represented as half-space constraints for motion planning. A curious fact: in a 2D plane, a half-space is simply a half-plane, but the term 'half-space' is used for any dimension, and in 1D, a half-space is a ray. The concept also appears in the theory of partial differential equations, where the 'half-space problem' refers to boundary value problems on a half-space domain, such as the heat equation on a semi-infinite rod.⁴

Hyperplane

An (n−1)-dimensional flat in an n-dimensional space that divides the space into two half-spaces.

Convex set

A set that contains the line segment between any two of its points.

Linear inequality

An inequality of the form a·x ≤ b, defining a half-space.

Polyhedron

The intersection of finitely many half-spaces.

Supporting hyperplane

A hyperplane that touches a convex set at its boundary and has the set entirely on one side.