Addition of sets
Vector addition and certain other (associative and commutative) operations on sets . The most important case is when the are convex sets in a Euclidean space .
The vector sum (with coefficients ) is defined in a linear space by the rule
where the are real numbers (see ). In the space , the vector sum is called also the Minkowski sum. The dependence of the volume on the is connected with mixed-volume theory. For convex , addition preserves convexity and reduces to addition of support functions (cf. Support function), while for -smooth strictly-convex , 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 ).
Firey -sums are defined in the class of convex bodies containing zero. When , the support function of the -sum is defined as , where are the support functions of the summands. For one carries out -addition of the corresponding polar bodies and takes the polar of the result (see ). Firey -sums are continuous with respect to and . The projection of a -sum onto a subspace is the -sum of the projections. When , the -sum coincides with the vector sum, when it is called the inverse sum (see ), when it gives the convex hull of the summands, and when it gives their intersection. For these four values, the -sum of polyhedra is a polyhedron, and when , the -sum of ellipsoids is an ellipsoid (see ).
The Blaschke sum is defined for convex bodies considered up to translation. It is defined by the addition of the area functions .
The sum along a subspace is defined in a vector space which is decomposed into the direct sum of two subspaces and . The sum of along is defined as
where is the translate of for which (see ).
|||R.T. Rockafellar, "Convex analysis" , Princeton Univ. Press (1970)|
|||W.J. Firey, "Some applications of means of convex bodies" Pacif. J. Math. , 14 (1964) pp. 53–60|
|||W.J. Firey, "Blaschke sums of convex bodies and mixed bodies" , Proc. Coll. Convexity (Copenhagen, 1965) , Copenhagen Univ. Mat. Inst. (1967) pp. 94–101|
|||A. Dinghas, "Minkowskische Summen und Integrale. Superadditive Mengenfunktionale. Isoperimetrische Ungleichungen" , Paris (1961)|
Addition of sets. V.P. Fedotov (originator), Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Addition_of_sets&oldid=15847