| # Half-Spaces LP and QP > Half-Spaces | Polyhedra | Standard Forms | Polyhedral Functions | Applications Definition Geometry Link with linear functions ## Definition A half-space is a set defined by a single affine inequality. Precisely, a half-space in is a set of the form where , . A half-space is a convex set, the boundary of which is a hyperplane. A half-space separates the whole space in two halves. The complement of the half-space is the open half-space .
| When , the half-space is the set of points which form an obtuse angle (between and ) with the vector . The boundary of this set is a subspace, the hyperplane of vectors orthogonal to .
| When , the corresponding half-space can be written as where is chosen such that . For example, is such a point on the boundary of the half-space (this particular choice corresponds to the minimum-norm solution to the equation ). Thus, the half-space above corresponds to the set of points such that (shown in dotted) forms an obtuse angle with the vector . The vector points outwards from the boundary.
Example: A half-space in . ## Link with linear functions Hyperplanes correspond to level sets of linear functions. Half-spaces represent sub-level sets of linear functions: the half-space above describes the set of points such that the linear function achieves the value , or less. A quick way to check which half of the space the half-space describe, is to look at where the origin is: if , then is in the half-space.