Minkowski addition Shapley–Folkman lemma

minkowski addition of sets. sum of squares q1=[0,1] and q2=[1,2] square q1+q2=[1,3].

in real vector space, minkowski sum of 2 (non-empty) sets q1 and q2 defined set q1 + q2 formed addition of vectors element-wise summand sets

q1 + q2 = { q1 + q2 : q1 ∈ q1 , q2 ∈ q2 }.

for example

{0, 1} + {0, 1} = {0+0, 0+1, 1+0, 1+1} = {0, 1, 2}.

by principle of mathematical induction, minkowski sum of finite family of (non-empty) sets

{qn : qn ≠ Ø , 1 ≤ n ≤ n }

is set formed element-wise addition of vectors

∑ qn = {∑ qn : qn ∈ qn}.




^ schneider (1993, p. xi) , rockafellar (1997, p. 16)
^ cite error: named reference carter94 invoked never defined (see page).
^ rockafellar (1997, p. 17) , starr (1997, p. 78)