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A mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110360/a1103601.png" /> associated with a set-valued function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110360/a1103602.png" /> from an (Abelian) [[Semi-group|semi-group]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110360/a1103603.png" /> to subsets of an (Abelian) semi-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110360/a1103604.png" /> which is a [[Homomorphism|homomorphism]] (of semi-groups) and a selection of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110360/a1103605.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110360/a1103606.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110360/a1103607.png" /> is the identity transformation on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110360/a1103608.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110360/a1103609.png" /> is said to be an additive selection on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110360/a11036010.png" />. An archetypical example of an additive selection is the mapping which subordinates to each non-empty compact set in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110360/a11036011.png" /> its maximal element. In <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110360/a11036012.png" />, similar selections can be defined by means of lexicographic orders, see [[#References|[a1]]] and [[Lexicographic order|Lexicographic order]]). They are Borel measurable but not continuous with respect to the [[Hausdorff metric|Hausdorff metric]] (for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110360/a11036013.png" />). A Lipschitz-continuous additive selection on the family of convex bodies in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110360/a11036014.png" /> is given by associating with each [[Convex body|convex body]] its Steiner point, see [[#References|[a2]]]. No such selections can exist in infinite dimensions, see [[#References|[a3]]], [[#References|[a4]]].
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A mapping  $  s : H \rightarrow G $
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associated with a set-valued function $  F $
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from an (Abelian) [[Semi-group|semi-group]] $  H $
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to subsets of an (Abelian) semi-group $  G $
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which is a [[Homomorphism|homomorphism]] (of semi-groups) and a selection of $  F $.  
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If $  G \subset  2  ^ {H} $
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and $  F $
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is the identity transformation on $  G $,  
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then $  s $
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is said to be an additive selection on $  G $.  
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An archetypical example of an additive selection is the mapping which subordinates to each non-empty compact set in $  \mathbf R $
 +
its maximal element. In $  \mathbf R  ^ {n} $,  
 +
similar selections can be defined by means of lexicographic orders, see [[#References|[a1]]] and [[Lexicographic order|Lexicographic order]]). They are Borel measurable but not continuous with respect to the [[Hausdorff metric|Hausdorff metric]] (for $  n > 1 $).  
 +
A Lipschitz-continuous additive selection on the family of convex bodies in $  \mathbf R  ^ {n} $
 +
is given by associating with each [[Convex body|convex body]] its Steiner point, see [[#References|[a2]]]. No such selections can exist in infinite dimensions, see [[#References|[a3]]], [[#References|[a4]]].
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  R. Živaljević,  "Extremal Minkowski additive selections of compact convex sets"  ''Proc. Amer. Math. Soc.'' , '''105'''  (1989)  pp. 697–700</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  R. Schneider,  "Convex bodies: the Brunn–Minkowski theory" , Cambridge Univ. Press  (1993)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  R.A. Vitale,  "The Steiner point in infinite dimensions"  ''Israel J. Math.'' , '''52'''  (1985)  pp. 245–250</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  K. Przesławski,  D. Yost,  "Continuity properties of selectors and Michael's theorem"  ''Michigan Math. J.'' , '''36'''  (1989)  pp. 113–134</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  R. Živaljević,  "Extremal Minkowski additive selections of compact convex sets"  ''Proc. Amer. Math. Soc.'' , '''105'''  (1989)  pp. 697–700</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  R. Schneider,  "Convex bodies: the Brunn–Minkowski theory" , Cambridge Univ. Press  (1993)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  R.A. Vitale,  "The Steiner point in infinite dimensions"  ''Israel J. Math.'' , '''52'''  (1985)  pp. 245–250</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  K. Przesławski,  D. Yost,  "Continuity properties of selectors and Michael's theorem"  ''Michigan Math. J.'' , '''36'''  (1989)  pp. 113–134</TD></TR></table>

Latest revision as of 16:09, 1 April 2020


A mapping $ s : H \rightarrow G $ associated with a set-valued function $ F $ from an (Abelian) semi-group $ H $ to subsets of an (Abelian) semi-group $ G $ which is a homomorphism (of semi-groups) and a selection of $ F $. If $ G \subset 2 ^ {H} $ and $ F $ is the identity transformation on $ G $, then $ s $ is said to be an additive selection on $ G $. An archetypical example of an additive selection is the mapping which subordinates to each non-empty compact set in $ \mathbf R $ its maximal element. In $ \mathbf R ^ {n} $, similar selections can be defined by means of lexicographic orders, see [a1] and Lexicographic order). They are Borel measurable but not continuous with respect to the Hausdorff metric (for $ n > 1 $). A Lipschitz-continuous additive selection on the family of convex bodies in $ \mathbf R ^ {n} $ is given by associating with each convex body its Steiner point, see [a2]. No such selections can exist in infinite dimensions, see [a3], [a4].

References

[a1] R. Živaljević, "Extremal Minkowski additive selections of compact convex sets" Proc. Amer. Math. Soc. , 105 (1989) pp. 697–700
[a2] R. Schneider, "Convex bodies: the Brunn–Minkowski theory" , Cambridge Univ. Press (1993)
[a3] R.A. Vitale, "The Steiner point in infinite dimensions" Israel J. Math. , 52 (1985) pp. 245–250
[a4] K. Przesławski, D. Yost, "Continuity properties of selectors and Michael's theorem" Michigan Math. J. , 36 (1989) pp. 113–134
How to Cite This Entry:
Additive selection. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Additive_selection&oldid=45029
This article was adapted from an original article by K. Przesławski (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article