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)