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 n-dimensional 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.
The open (closed) upper half-space is the half-space of all <math> x_1, x_2, \dots, x_n </math> such that <math> x_n \ge 0 </math> (<math> x_n > 0 </math>). The open (closed) lower half-space is defined similarly, by requiring that <math> x_n </math> be negative (non-positive).
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:
<math display="block"> a_1x_1+a_2x_2+\cdots+a_nx_n > b. </math>
A non-strict one specifies a closed half-space:
<math display="block"> a_1x_1+a_2x_2+\cdots+a_nx_n \geq b. </math>
Here, one assumes that not all of the real numbers a<sub>1</sub>, a<sub>2</sub>, ..., a<sub>n</sub> are zero.
A half-space is a convex set.
See also
- Hemisphere (geometry)
- Line (geometry)
- Nef polygon, construction of polyhedra using half-spaces
- Poincaré half-plane model
- Quadrant (solid geometry)
- Siegel upper half-space