Half-space (geometry)

{{Short description|Bisection of Euclidean space by a hyperplane}}{{More citations needed|date=December 2024}}

In geometry, a half-space is either of the two parts into which a plane divides the three-dimensional Euclidean space.{{Cite Merriam-Webster|half-space}} If the space is two-dimensional, then a half-space is called a half-plane (open or closed).{{Cite web |last=Weisstein |first=Eric W. |title=Half-Space |url=https://mathworld.wolfram.com/Half-Space.html |access-date=2024-12-04 |website=Wolfram MathWorld |language=en}}{{Cite web |last=Weisstein |first=Eric W. |title=Half-Plane |url=https://mathworld.wolfram.com/Half-Plane.html |access-date=2024-12-04 |website=Wolfram MathWorld |language=en}} A half-space in a one-dimensional space is called a half-line{{Cite Merriam-Webster|half line}} or ray.

More generally, a half-space is either of the two parts into which a hyperplane divides an n-dimensional 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 (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).

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:

:$a\_1x\_1+a\_2x\_2+\backslash cdots+a\_nx\_n>b$

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

:$a\_1x\_1+a\_2x\_2+\backslash cdots+a\_nx\_n\backslash geq\; b$

Here, one assumes that not all of the real numbers a₁, a₂, ..., a_n are zero.

A half-space is a convex set.