Half-space

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For other uses, see Half-space (disambiguation).

The red plane determines the blue half-space.

In geometry, a half-space is either of the two parts into which a plane divides the three-dimensional space. More generally, a half-space is either of the two parts into which a hyperplane divides an affine space.

One can have open and closed half-spaces. 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

a₁x₁ + a₂x₂ + ... + a_nx_n > b

specifies an open half-space, while a non-strict one

a₁x₁ + a₂x₂ + ... + a_nx_n $\geq$ b

specifies a closed half-space. Here, one assumes that not all of the real numbers a₁, a₂, ..., a_n are zero.

[edit] Properties

A half-space is a convex set.
Any convex set can be described as the (possibly infinite) intersection of halfspaces

[edit] Upper and lower half-spaces

The open (closed) upper half-space is the half-space of all (x₁, x₂, ..., 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).