Addition of sets

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)