halfspace

A half-space is a fundamental concept in Euclidean geometry and convex analysis. In n-dimensional space R^n, a half-space is one side of a hyperplane. Given a nonzero vector a ∈ R^n and a scalar b ∈ R, the set {x ∈ R^n : a^T x ≤ b} is a closed half-space, and its boundary is the hyperplane {x ∈ R^n : a^T x = b}. The set {x ∈ R^n : a^T x < b} is an open half-space, which does not include the boundary. Replacing ≤ or < with ≥ or > yields the opposite half-space.

Half-spaces are convex and unbounded. The boundary hyperplane a^T x = b has codimension one. Any finite intersection of half-spaces is a convex polyhedron, which may be unbounded. This representation, known as the H-representation, is central to linear programming and convex analysis. By contrast, a polyhedron can also be described by its vertices and rays (V-representation).

Examples in R^2 include the half-plane x ≥ 0, which consists of all points to the right of the y-axis and includes the axis itself; its boundary is the line x = 0. The open half-plane x > 0 excludes the boundary. These ideas generalize to higher dimensions, with half-spaces defined by linear inequalities.

Applications of half-spaces include formulating feasible regions in linear programming, proving separation theorems in convex analysis, and serving as basic building blocks in computational geometry and related fields.