Wikipedia
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:
ax + ax + ⋯ + ax > b
A non-strict one specifies a closed half-space:
ax + ax + ⋯ + ax ≥ b
Here, one assumes that not all of the real numbers a, a, ..., a are zero.
Half-space may refer to:
- Half-space (geometry), either of the two parts into which a plane divides Euclidean space
- Half-space (punctuation), a spacing character half the width of a regular space
- Half-space model, a model of the hyperbolic plane or higher-dimensional hyperbolic spaces using a Euclidean half-space
- Half-space model (oceanography), an estimate for seabed height in areas without significant subduction
- Siegel upper half-space, a set of complex matrices with positive definite imaginary part