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Prove that the intersection of a number of finite convex sets is also a convex set

  • Thread starter retspool
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Homework Statement



Prove that the intersection of a number of finite convex sets is also a convex set

Homework Equations



I have a set is convex if there exists x, y in the convex S then

f(ax + (1-a)y< af(x) + (1-a)y

where 0<a<1


The Attempt at a Solution



i can prove that
f(ax + (1-a)y) < f(x) given that x is a global minimizer

then i guess that i could find another arbritary point close to x , x_1, x_2 and add their given function satisfying the convex condition to get

Sum f(axi + (1-a)y) < Sumf(xi) where i= 1, 2,...n


any help would be appreciated
 

Answers and Replies

  • #2
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Your definition of "convex set" is wrong; there is no function involved. A set [tex]S[/tex] (in some real vector space [tex]V[/tex]) is convex if, whenever [tex]x, y \in S[/tex] and [tex]0 \leq a \leq 1[/tex], then also [tex]ax + (1 - a)y \in S[/tex].

Once you correct that, if you find yourself working too hard, you're doing something wrong. Just chase the definitions.
 

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