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Half-space (geometry)

Bisection of Euclidean space by a hyperplane

Wikipedia contributorsAdapted for Fathomly · View authors
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Source updated April 28, 2026

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 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 x_{1},x_{2},\dots ,x_{n} such that x_{n}\geq 0 (x_{n}>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_{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 a1, a2, ..., an are zero.

A half-space is a convex set.

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This article is adapted from the Wikipedia article “Half-space (geometry)”, written by its contributors and licensed under CC BY-SA 4.0. Fathomly has changed the layout, removed citation markers, navigation and maintenance notices, and adjusted punctuation. This adapted version is shared under the same license. For references, see the original article.

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