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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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The author is known as Wilber Pegues. Since I was eighteen I've been operating as a bookkeeper but quickly my spouse and I will start our personal company. The preferred hobby for him and his kids is to perform lacross and he would never give it up. For many years he's been residing in Alaska and he doesn't plan on altering it.
My homepage ... free psychic readings, http://Vjchary.com/success/?author=17500,