Lemma of Shapley and Folkman Shapley–Folkman lemma

a winner of 2012 nobel award in economics, lloyd shapley proved shapley–folkman lemma jon folkman.

for representation of point x, shapley–folkman lemma states if dimension d less number of summands

d < n

then convexification needed only d summand-sets, choice depends on x: point has representation

x
=

∑

1
≤

d

≤

d





q

d



+

∑

d
+
1
≤

n

≤

n





q

n





{\displaystyle x=\sum _{1\leq {d}\leq {d}}{q_{d}}+\sum _{d+1\leq {n}\leq {n}}{q_{n}}}

where qd belongs convex hull of qd for d (or fewer) summand-sets and qn belongs to qn remaining sets. that is,

x
∈


∑

1
≤

d

≤

d




conv
⁡

(

q

d


)


+

∑

d
+
1
≤

n

≤

n





q

n






{\displaystyle x\in {\sum _{1\leq {d}\leq {d}}{\operatorname {conv} {(q_{d})}}+\sum _{d+1\leq {n}\leq {n}}{q_{n}}}}

for re-indexing of summand sets; re-indexing depends on particular point x being represented.

the shapley–folkman lemma implies, example, every point in [0, 2] sum of integer from {0, 1} , real number from [0, 1].

dimension of real vector space

conversely, shapley–folkman lemma characterizes dimension of finite-dimensional, real vector spaces. is, if vector space obeys shapley–folkman lemma natural number d, , no number less than d, dimension exactly d; shapley–folkman lemma holds finite-dimensional vector spaces.