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| In [[mathematics]], the '''affine hull''' of a [[set (mathematics)|set]] ''S'' in [[Euclidean space]] '''R'''<sup>''n''</sup> is the smallest [[affine set]] containing ''S'', or equivalently, the [[intersection (set theory)|intersection]] of all affine sets containing ''S''. Here, an ''affine set'' may be defined as the [[translation (mathematics)|translation]] of a [[vector subspace]].
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| The affine hull aff(''S'') of ''S'' is the set of all [[affine combination]]s of elements of ''S'', that is,
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| :<math>\operatorname{aff} (S)=\left\{\sum_{i=1}^k \alpha_i x_i \Bigg | k>0, \, x_i\in S, \, \alpha_i\in \mathbb{R}, \, \sum_{i=1}^k \alpha_i=1 \right\}.</math>
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| ==Examples==
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| *The affine hull of a set of two different points is the line through them.
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| *The affine hull of a set of three points not on one line is the plane going through them.
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| *The affine hull of a set of four points not in a plane in '''R'''<sup>''3''</sup> is the entire space '''R'''<sup>''3''</sup>.
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| ==Properties==
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| * <math>\mathrm{aff}(\mathrm{aff}(S)) = \mathrm{aff}(S)</math>
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| * <math>\mathrm{aff}(S)</math> is a [[closed set]]
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| ==Related sets==
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| *If instead of an affine combination one uses a [[convex combination]], that is one requires in the formula above that all <math>\alpha_i</math> be non-negative, one obtains the [[convex hull]] of ''S'', which cannot be larger than the affine hull of ''S'' as more restrictions are involved.
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| *The notion of [[conical combination]] gives rise to the notion of the [[conical hull]]
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| *If however one puts no restrictions at all on the numbers <math>\alpha_i</math>, instead of an affine combination one has a [[linear combination]], and the resulting set is the [[linear span]] of ''S'', which contains the affine hull of ''S''.
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| ==References==
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| * R.J. Webster, ''Convexity'', Oxford University Press, 1994. ISBN 0-19-853147-8.
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| [[Category:Affine geometry]]
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| [[Category:Closure operators]]
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Hi there, I am Alyson Boon even though it is not the title on my birth certificate. She works as a travel agent but soon she'll be on her own. Mississippi is the only place I've been residing in but I will have to transfer in a yr or two. To play lacross is one of the things she enjoys most.
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