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
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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
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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) |
Addition of sets. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Addition_of_sets&oldid=45025