Bisection of Euclidean space by a hyperplane

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.

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:

- $a_{1}x_{1}+a_{2}x_{2}+\cdots +a_{n}x_{n}>b$

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

- $a_{1}x_{1}+a_{2}x_{2}+\cdots +a_{n}x_{n}\geq b$

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

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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).