Convex sets Shapley–Folkman lemma

in real vector space, non-empty set q defined convex if, each pair of points, every point on line segment joins them subset of q. example, solid disk 



∙


{\displaystyle \bullet }

convex circle 



∘


{\displaystyle \circ }

not, because not contain line segment joining points 



⊘


{\displaystyle \oslash }

; non-convex set of 3 integers {0, 1, 2} contained in interval [0, 2], convex. example, solid cube convex; however, hollow or dented, example, crescent shape, non-convex. empty set convex, either definition or vacuously, depending on author.

more formally, set q convex if, points v0 and v1 in q , every real number λ in unit interval [0,1], point

(1 − λ) v0 + λv1

is member of q.

by mathematical induction, set q convex if , only if every convex combination of members of q belongs to q. definition, convex combination of indexed subset {v0, v1, . . . , vd} of vector space weighted average λ0v0 + λ1v1 + . . . + λdvd, indexed set of non-negative real numbers {λd} satisfying equation λ0 + λ1 + . . .  + λd = 1.

the definition of convex set implies intersection of 2 convex sets convex set. more generally, intersection of family of convex sets convex set. in particular, intersection of 2 disjoint sets empty set, convex.