{{Short description|Bisection of Euclidean space by a hyperplane}}{{Expand language|topic=|langcode=ru|date=April 2026}} {{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.<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=4 December 2024 |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=4 December 2024 |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''.''
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.{{r|mg}}
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 <math> x_1, x_2, \dots, x_n </math> such that <math> x_n \ge 0 </math> (<math> x_n > 0 </math>). The open (closed) ''lower half-space'' is defined similarly, by requiring that <math> x_n </math> 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: <math display="block"> a_1x_1+a_2x_2+\cdots+a_nx_n > b. </math> A non-strict one specifies a closed half-space: <math display="block"> 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.
==See also== * Hemisphere (geometry) * Line (geometry) * Nef polygon, construction of polyhedra using half-spaces * Poincaré half-plane model * Quadrant (solid geometry) * Siegel upper half-space
==References== {{Reflist|refs=
<ref name=mg>{{cite book | title = Understanding and Using Linear Programming | first1 = Jiri | last1 = Matousek | first2 = Bernd | last2 = Gärtner | date = 4 July 2007 | url = https://books.google.com/books?id=6MO_RS4z0w8C&pg=PA51 | page = 51 | publisher = Springer | isbn = 978-3-540-30717-4 }}</ref>
}}
==External links== * {{springer|title=Half-plane|id=p/h046170}} * {{Mathworld | urlname=Half-Space | title=Half-Space }}
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{{DEFAULTSORT:Half-Space}} Category:Euclidean geometry