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 [1]). 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 [4]).

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 [2]). 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 [1]), 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 [2]).

The Blaschke sum is defined for convex bodies 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 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 [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)