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| In [[geometry]], a '''half-space''' is either of the two parts into which a [[plane (geometry)|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 [[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.
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| 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.
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| 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 '''[[Line_(mathematics)#Ray|ray]]'''.
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| A half-space may be specified by a linear inequality, derived from the [[linear equation]] that specifies the defining hyperplane.
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| A strict linear [[inequality (mathematics)|inequality]] specifies an open half-space:
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| :<math>a_1x_1+a_2x_2+\cdots+a_nx_n>b</math>
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| A non-strict one specifies a closed half-space:
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| :<math>a_1x_1+a_2x_2+\cdots+a_nx_n\geq b</math>
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| Here, one assumes that not all of the real numbers ''a''<sub>1</sub>, ''a''<sub>2</sub>, ..., ''a''<sub>''n''</sub> are zero. | |
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| ==Properties==
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| * A half-space is a [[convex set]].
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| * Any [[convex set]] can be described as the (possibly infinite) intersection of half-spaces.
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| ==Upper and lower half-spaces==
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| 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).
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| ==See also==
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| * [[Half-line]]
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| * [[Upper half-plane]]
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| * [[Poincaré half-plane model]]
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| * [[Siegel upper half-space]]
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| * [[Nef polygon]] , construction of [[polyhedra]] using half-spaces.
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| ==External links==
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| * {{Mathworld | urlname=Half-Space | title=Half-Space }}
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| {{DEFAULTSORT:Half-Space}}
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| [[Category:Euclidean geometry]]
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I am Angeline and was born on 21 October 1989. My hobbies are People watching and Fantasy Football.
Here is my weblog super easy forex trade