Euclidean geometry

Half-space (geometry)

In geometry, a half-space is either of the two parts into which a plane divides the three-dimensional Euclidean space. If the space is two-dimensional, then a half-space is called a half-plane (open or closed). A half-space in a one-dimensional space is called a half-line or ray. More generally, a half-space is either of the two parts into which a hyperplane divides an affine space. That is, the points that are not incident to the hyperplane are partitioned into two convex sets (i.e., half-spaces), such that any subspace connecting a point in one set to a point in the other must intersect the hyperplane. A half-space can be either open or closed. An open half-space is either of the two open sets produced by the subtraction of a hyperplane from the affine space. A closed half-space is the union of an open half-space and the hyperplane that defines it. A half-space may be specified by a linear inequality, derived from the linear equation that specifies the defining hyperplane.A strict linear inequality specifies an open half-space: A non-strict one specifies a closed half-space: Here, one assumes that not all of the real numbers a1, a2, ..., an are zero. (Wikipedia).

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Linear equation | Upper half-plane | Line (geometry) | Affine space | Hyperplane | Inequality (mathematics) | Nef polygon | Poincaré half-plane model | Geometry | Euclidean space | Plane (geometry) | Siegel upper half-space | Convex set | Open set