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Vector addition and certain other (associative and commutative) operations on sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010600/a0106001.png" />. The most important case is when the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010600/a0106002.png" /> are convex sets in a Euclidean space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010600/a0106003.png" />.
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The vector sum (with coefficients <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010600/a0106004.png" />) is defined in a linear space by the rule
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010600/a0106005.png" /></td> </tr></table>
+
Vector addition and certain other (associative and commutative) operations on sets  $  A _ {i} $.  
 +
The most important case is when the  $  A _ {i} $
 +
are convex sets in a Euclidean space  $  \mathbf R  ^ {n} $.
  
where the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010600/a0106006.png" /> are real numbers (see [[#References|[1]]]). In the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010600/a0106007.png" />, the vector sum is called also the Minkowski sum. The dependence of the volume <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010600/a0106008.png" /> on the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010600/a0106009.png" /> is connected with [[Mixed-volume theory|mixed-volume theory]]. For convex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010600/a01060010.png" />, addition preserves convexity and reduces to addition of support functions (cf. [[Support function|Support function]]), while for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010600/a01060011.png" />-smooth strictly-convex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010600/a01060012.png" />, it is characterized by the addition of the mean values of the radii of curvature at points with a common normal.
+
The vector sum (with coefficients  $  \lambda _ {i} $)
 +
is defined in a linear space by the rule
 +
 
 +
$$
 +
= \sum _ { i } \lambda _ {i} A _ {i}  = \
 +
\cup _ {x _ {i} \in A _ {i} }
 +
\left \{ \sum _ { i } \lambda _ {i} x _ {i} \right \} .
 +
$$
 +
 
 +
where the  $  \lambda _ {i} $
 +
are real numbers (see [[#References|[1]]]). In the space $  \mathbf R  ^ {n} $,  
 +
the vector sum is called also the Minkowski sum. The dependence of the volume $  S $
 +
on the $  \lambda _ {i} $
 +
is connected with [[Mixed-volume theory|mixed-volume theory]]. For convex $  A _ {i} $,  
 +
addition preserves convexity and reduces to addition of support functions (cf. [[Support function|Support function]]), while for $  C  ^ {2} $-
 +
smooth strictly-convex $  A _ {i} \subset  \mathbf R  ^ {n} $,  
 +
it is characterized by the addition of the mean values of the radii of curvature at points with a common normal.
  
 
Further examples are addition of sets up to translation, addition of closed sets (along with closure of the result, see [[Convex sets, linear space of|Convex sets, linear space of]]; [[Convex sets, metric space of|Convex sets, metric space of]]), integration of a continual family of sets, and addition in commutative semi-groups (see [[#References|[4]]]).
 
Further examples are addition of sets up to translation, addition of closed sets (along with closure of the result, see [[Convex sets, linear space of|Convex sets, linear space of]]; [[Convex sets, metric space of|Convex sets, metric space of]]), integration of a continual family of sets, and addition in commutative semi-groups (see [[#References|[4]]]).
  
Firey <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010600/a01060014.png" />-sums are defined in the class of convex bodies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010600/a01060015.png" /> containing zero. When <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010600/a01060016.png" />, the support function of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010600/a01060017.png" />-sum is defined as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010600/a01060018.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010600/a01060019.png" /> are the support functions of the summands. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010600/a01060020.png" /> one carries out <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010600/a01060021.png" />-addition of the corresponding polar bodies and takes the polar of the result (see [[#References|[2]]]). Firey <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010600/a01060022.png" />-sums are continuous with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010600/a01060023.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010600/a01060024.png" />. The projection of a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010600/a01060025.png" />-sum onto a subspace is the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010600/a01060026.png" />-sum of the projections. When <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010600/a01060027.png" />, the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010600/a01060028.png" />-sum coincides with the vector sum, when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010600/a01060029.png" /> it is called the inverse sum (see [[#References|[1]]]), when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010600/a01060030.png" /> it gives the convex hull of the summands, and when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010600/a01060031.png" /> it gives their intersection. For these four values, the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010600/a01060032.png" />-sum of polyhedra is a polyhedron, and when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010600/a01060033.png" />, the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010600/a01060034.png" />-sum of ellipsoids is an ellipsoid (see [[#References|[2]]]).
+
Firey $  p $-
 +
sums are defined in the class of convex bodies $  A _ {i} \subset  \mathbf R  ^ {n} $
 +
containing zero. When $  p \geq  1 $,  
 +
the support function of the $  p $-
 +
sum is defined as $  ( \sum _ {i} H _ {i}  ^ {p} ) ^ {1/p } $,  
 +
where $  H _ {i} $
 +
are the support functions of the summands. For $  p \leq  -1 $
 +
one carries out $  ( -p ) $-
 +
addition of the corresponding polar bodies and takes the polar of the result (see [[#References|[2]]]). Firey $  p $-
 +
sums are continuous with respect to $  A _ {i} $
 +
and $  p $.  
 +
The projection of a $  p $-
 +
sum onto a subspace is the $  p $-
 +
sum of the projections. When $  p = 1 $,  
 +
the $  p $-
 +
sum coincides with the vector sum, when $  p = -1 $
 +
it is called the inverse sum (see [[#References|[1]]]), when $  p = + \infty $
 +
it gives the convex hull of the summands, and when $  p = - \infty $
 +
it gives their intersection. For these four values, the $  p $-
 +
sum of polyhedra is a polyhedron, and when $  p = \pm 2 $,  
 +
the $  p $-
 +
sum of ellipsoids is an ellipsoid (see [[#References|[2]]]).
  
The Blaschke sum is defined for convex bodies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010600/a01060035.png" /> considered up to translation. It is defined by the addition of the area functions [[#References|[3]]].
+
The Blaschke sum is defined for convex bodies $  A _ {i} \subset  \mathbf R  ^ {n} $
 +
considered up to translation. It is defined by the addition of the area functions [[#References|[3]]].
  
The sum along a subspace is defined in a vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010600/a01060036.png" /> which is decomposed into the direct sum of two subspaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010600/a01060037.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010600/a01060038.png" />. The sum of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010600/a01060039.png" /> along <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010600/a01060040.png" /> is defined as
+
The sum along a subspace is defined in a vector space $  X $
 +
which is decomposed into the direct sum of two subspaces $  Y $
 +
and $  Z $.  
 +
The sum of $  A _ {i} $
 +
along $  Y $
 +
is defined as
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010600/a01060041.png" /></td> </tr></table>
+
$$
 +
\cup _ {z \subset  Z }
 +
\left \{ \sum _ { i } ( Y _ {z} \cap A _ {i} ) \right \} ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010600/a01060042.png" /> is the translate of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010600/a01060043.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010600/a01060044.png" /> (see [[#References|[1]]]).
+
where $  Y _ {z} $
 +
is the translate of $  Y $
 +
for which $  Y _ {z} \cap Z = \{ z \} $(
 +
see [[#References|[1]]]).
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  R.T. Rockafellar,  "Convex analysis" , Princeton Univ. Press  (1970)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  W.J. Firey,  "Some applications of means of convex bodies"  ''Pacif. J. Math.'' , '''14'''  (1964)  pp. 53–60</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  W.J. Firey,  "Blaschke sums of convex bodies and mixed bodies" , ''Proc. Coll. Convexity (Copenhagen, 1965)'' , Copenhagen Univ. Mat. Inst.  (1967)  pp. 94–101</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  A. Dinghas,  "Minkowskische Summen und Integrale. Superadditive Mengenfunktionale. Isoperimetrische Ungleichungen" , Paris  (1961)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  R.T. Rockafellar,  "Convex analysis" , Princeton Univ. Press  (1970)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  W.J. Firey,  "Some applications of means of convex bodies"  ''Pacif. J. Math.'' , '''14'''  (1964)  pp. 53–60</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  W.J. Firey,  "Blaschke sums of convex bodies and mixed bodies" , ''Proc. Coll. Convexity (Copenhagen, 1965)'' , Copenhagen Univ. Mat. Inst.  (1967)  pp. 94–101</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  A. Dinghas,  "Minkowskische Summen und Integrale. Superadditive Mengenfunktionale. Isoperimetrische Ungleichungen" , Paris  (1961)</TD></TR></table>

Latest revision as of 16:09, 1 April 2020


Vector addition and certain other (associative and commutative) operations on sets $ A _ {i} $. The most important case is when the $ A _ {i} $ are convex sets in a Euclidean space $ \mathbf R ^ {n} $.

The vector sum (with coefficients $ \lambda _ {i} $) is defined in a linear space by the rule

$$ S = \sum _ { i } \lambda _ {i} A _ {i} = \ \cup _ {x _ {i} \in A _ {i} } \left \{ \sum _ { i } \lambda _ {i} x _ {i} \right \} . $$

where the $ \lambda _ {i} $ are real numbers (see [1]). In the space $ \mathbf R ^ {n} $, the vector sum is called also the Minkowski sum. The dependence of the volume $ S $ on the $ \lambda _ {i} $ is connected with mixed-volume theory. For convex $ A _ {i} $, addition preserves convexity and reduces to addition of support functions (cf. Support function), while for $ C ^ {2} $- smooth strictly-convex $ A _ {i} \subset \mathbf R ^ {n} $, it is characterized by the addition of the mean values of the radii of curvature at points with a common normal.

Further examples are addition of sets up to translation, addition of closed sets (along with closure of the result, see Convex sets, linear space of; Convex sets, metric space of), integration of a continual family of sets, and addition in commutative semi-groups (see [4]).

Firey $ p $- sums are defined in the class of convex bodies $ A _ {i} \subset \mathbf R ^ {n} $ containing zero. When $ p \geq 1 $, the support function of the $ p $- sum is defined as $ ( \sum _ {i} H _ {i} ^ {p} ) ^ {1/p } $, where $ H _ {i} $ are the support functions of the summands. For $ p \leq -1 $ one carries out $ ( -p ) $- addition of the corresponding polar bodies and takes the polar of the result (see [2]). Firey $ p $- sums are continuous with respect to $ A _ {i} $ and $ p $. The projection of a $ p $- sum onto a subspace is the $ p $- sum of the projections. When $ p = 1 $, the $ p $- sum coincides with the vector sum, when $ p = -1 $ it is called the inverse sum (see [1]), when $ p = + \infty $ it gives the convex hull of the summands, and when $ p = - \infty $ it gives their intersection. For these four values, the $ p $- sum of polyhedra is a polyhedron, and when $ p = \pm 2 $, the $ p $- sum of ellipsoids is an ellipsoid (see [2]).

The Blaschke sum is defined for convex bodies $ A _ {i} \subset \mathbf R ^ {n} $ considered up to translation. It is defined by the addition of the area functions [3].

The sum along a subspace is defined in a vector space $ X $ which is decomposed into the direct sum of two subspaces $ Y $ and $ Z $. The sum of $ A _ {i} $ along $ Y $ is defined as

$$ \cup _ {z \subset Z } \left \{ \sum _ { i } ( Y _ {z} \cap A _ {i} ) \right \} , $$

where $ Y _ {z} $ is the translate of $ Y $ for which $ Y _ {z} \cap Z = \{ z \} $( see [1]).

References

[1] R.T. Rockafellar, "Convex analysis" , Princeton Univ. Press (1970)
[2] W.J. Firey, "Some applications of means of convex bodies" Pacif. J. Math. , 14 (1964) pp. 53–60
[3] W.J. Firey, "Blaschke sums of convex bodies and mixed bodies" , Proc. Coll. Convexity (Copenhagen, 1965) , Copenhagen Univ. Mat. Inst. (1967) pp. 94–101
[4] A. Dinghas, "Minkowskische Summen und Integrale. Superadditive Mengenfunktionale. Isoperimetrische Ungleichungen" , Paris (1961)
How to Cite This Entry:
Addition of sets. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Addition_of_sets&oldid=45025
This article was adapted from an original article by V.P. Fedotov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article