Template:Short descriptionTemplate:More citations needed In geometry, a half-space is either of the two parts into which a plane divides the three-dimensional Euclidean space.<ref>Template:Cite Merriam-Webster</ref> If the space is two-dimensional, then a half-space is called a half-plane (open or closed).<ref name=":0">{{#invoke:citation/CS1|citation |CitationClass=web }}</ref><ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> A half-space in a one-dimensional space is called a half-line<ref>Template:Cite Merriam-Webster</ref> or ray.
More generally, a half-space is either of the two parts into which a hyperplane divides an n-dimensional space.<ref name=":0" /> 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 (x1, x2, ..., xn) such that xn > 0 (≥ 0). The open (closed) lower half-space is defined similarly, by requiring that xn 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>a_1x_1+a_2x_2+\cdots+a_nx_n>b</math>
A non-strict one specifies a closed half-space:
- <math>a_1x_1+a_2x_2+\cdots+a_nx_n\geq b</math>
Here, one assumes that not all of the real numbers a1, a2, ..., an are zero.
A half-space is a convex set.
See alsoEdit
- Hemisphere (geometry)
- Line (geometry)
- Poincaré half-plane model
- Siegel upper half-space
- Nef polygon, construction of polyhedra using half-spaces.