Half-space (geometry)
In geometry, a half-space is either of the two parts into which a plane divides the three-dimensional Euclidean space. 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.
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 ray.
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 a_{1}, a_{2}, ..., a_{n} are zero.
Properties
- A half-space is a convex set.
- Any convex set can be described as the (possibly infinite) intersection of half-spaces.
Upper and lower half-spaces
The open (closed) upper half-space is the half-space of all (x_{1}, x_{2}, ..., x_{n}) such that x_{n} > 0 (≥ 0). The open (closed) lower half-space is defined similarly, by requiring that x_{n} be negative (non-positive).
See also
- line (geometry)
- Upper half-plane
- Poincaré half-plane model
- Siegel upper half-space
- Nef polygon, construction of polyhedra using half-spaces.
External links
- Hazewinkel, Michiel, ed. (2001) [1994], "Half-plane", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4
- Weisstein, Eric W. "Half-Space". MathWorld.