Halbraum

Halbraum, in geometry and linear algebra, refers to a half-space: a subset of Euclidean space defined by a linear inequality. For a nonzero vector a in R^n and a real number b, the closed half-space is H = { x ∈ R^n | a^T x ≤ b }; the open half-space is { x ∈ R^n | a^T x < b }. The boundary of a half-space is the hyperplane a^T x = b.

Half-spaces are convex and unbounded. They are closed when defined with ≤ and open when defined with <; the boundary is the corresponding hyperplane. In simple terms, a half-space consists of all points on one side of a flat, infinite boundary.

Examples include the right half-plane in the plane, defined by x1 ≥ 0, whose boundary is the y-axis, and more generally any set { x | a^T x ≤ b } in higher dimensions. The geometry is characterized by a single orientation given by the normal vector a and a threshold b.

Applications and structure: The intersection of finitely many half-spaces is a convex polyhedron (or polytope if bounded). Conversely, many convex sets useful in optimization can be described as intersections of half-spaces, making Halbraum a fundamental building block in linear programming and convex optimization. The separation (or supporting hyperplane) theorem guarantees that a point outside a convex set can be separated by a hyperplane, yielding a half-space that isolates the point from the set.

Name note: Halbraum is the German term for half-space; in English-language mathematics, the term half-space is commonly used.