View source for Addition of sets

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" />.

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

<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>

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.

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]]]).

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 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

<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>

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]]]).

====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>