Editing Half-space (geometry) (section)

{{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 (geometry)|plane]] divides the three-dimensional [[Euclidean space]].<ref>{{Cite Merriam-Webster|half-space}}</ref> If the space is [[two-dimensional]], then a half-space is called a ''[[half-plane]]'' (open or closed).<ref name=":0">{{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}}</ref><ref>{{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}}</ref> A half-space in a [[one-dimensional]] space is called a ''half-line''<ref>{{Cite Merriam-Webster|half line}}</ref> or [[ray (mathematics)|ray]]''.''

More generally, a '''half-space''' is either of the two parts into which a [[hyperplane]] divides an n-dimensional [[space]].<ref name=":0" /> That is, the points that are not incident to the hyperplane are [[partition (set theory)|partitioned]] into two [[convex set]]s (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 set]]s 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''<sub>1</sub>, ''x''<sub>2</sub>, ..., ''x''<sub>''n''</sub>) such that ''x''<sub>''n''</sub> > 0 (≥ 0). The open (closed) ''lower half-space'' is defined similarly, by requiring that ''x''<sub>''n''</sub> 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 (mathematics)|inequality]] specifies an open half-space:
:<math>a_1x_1+a_2x_2+\cdots+a_nx_n>b</math>
A non-strict one specifies a closed half-space:
:<math>a_1x_1+a_2x_2+\cdots+a_nx_n\geq b</math>
Here, one assumes that not all of the real numbers ''a''<sub>1</sub>, ''a''<sub>2</sub>, ..., ''a''<sub>''n''</sub> are zero.

A half-space is a [[convex set]].