# Half-space (geometry)

__: Bisection of Euclidean space by a hyperplane__

**Short description**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.

The open (closed) *upper half-space* is the half-space of all (*x*_{1}, *x*_{2}, ..., *x*_{n}) such that *x*_{n} > 0 (â‰¥ 0). The open (closed) *lower half-space* is defined similarly, by requiring that *x*_{n} 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]\displaystyle{ a_1x_1+a_2x_2+\cdots+a_nx_n\gt b }[/math]

A non-strict one specifies a closed half-space:

- [math]\displaystyle{ a_1x_1+a_2x_2+\cdots+a_nx_n\geq b }[/math]

Here, one assumes that not all of the real numbers *a*_{1}, *a*_{2}, ..., *a*_{n} are zero.

A half-space is a convex set.

## See also

- Hemisphere (geometry)
- Line (geometry)
- PoincarĂ© half-plane model
- Siegel upper half-space
- Nef polygon, construction of polyhedra using half-spaces.

## External links

- Hazewinkel, Michiel, ed. (2001), "Half-plane",
*Encyclopedia of Mathematics*, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4, https://www.encyclopediaofmath.org/index.php?title=p/h046170 - Weisstein, Eric W.. "Half-Space". http://mathworld.wolfram.com/Half-Space.html.

Original source: https://en.wikipedia.org/wiki/Half-space (geometry).
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