Shapley–Folkman lemma

The Shapley–Folkman lemma is a result in convex geometry with applications in mathematical economics that describes the Minkowski addition of sets in a vector space. Minkowski addition is defined as the addition of the sets' members: for example, adding the set consisting of the integers zero and one to itself yields the set consisting of zero, one, and two:

The Shapley–Folkman lemma and related results provide an affirmative answer to the question, "Is the sum of many sets close to being convex?" A set is defined to be convex if every line segment joining two of its points is a subset in the set: For example, the solid disk ${\displaystyle \bullet }$ is a convex set but the circle ${\displaystyle \circ }$ is not, because the line segment joining two distinct points ${\displaystyle \oslash }$ is not a subset of the circle. The Shapley–Folkman lemma suggests that if the number of summed sets exceeds the dimension of the vector space, then their Minkowski sum is approximately convex.

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