# 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*.