# Half-space

Half-space

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.

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

$a_1x_1+a_2x_2+\cdots+a_nx_n>b$

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

$a_1x_1+a_2x_2+\cdots+a_nx_n\geq b$

specifies a closed half-space. Here, one assumes that not all of the real numbers a1, a2, ..., an 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 (x1, x2, ..., xn) such that xn > 0 (≥ 0). The open (closed) lower half-space is defined similarly, by requiring that xn be negative (non-positive).

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