# Convex set proof

• Feb 18th 2010, 07:52 AM
ROOZ1234
Convex set proof
I have a question about convex set. how can I prove that a set is convex if and only if its intersection with any line is convex. I have worked out proof but it is not that rigorous and I was wondering if someone could help me with this
• Feb 18th 2010, 08:06 AM
Plato
Quote:

Originally Posted by ROOZ1234
I have a question about convex set. how can I prove that a set is convex if and only if its intersection with any line is convex. I have worked out proof but it is not that rigorous and I was wondering if someone could help me with this

Why not post what you have done? We can look at it then.
Here is a hint: Two points determine a unique line and lines are convex sets.
• Feb 18th 2010, 08:32 AM
ROOZ1234
if S is convex
exist x,y that belongs to S -> lambda(x) + (1-lambda) (y) belongs to S
therefore x and y are on the line L
->lambda(x) + (1-lambda) (y) belongs to L
->lambda(x) + (1-lambda) (y) belongs to intersection of L and S
therefore S interesection L is convex

there exists x,y that belongs to S intersection with L -> lambda(x) + (1-lambda) (y) belongs to S intersection with L

-> lambda(x) + (1-lambda) (y) belong to S
->S is convex
• Feb 18th 2010, 09:02 AM
Plato
You certainly have all to components there.
I do not know the level of rigor required of you.
• Feb 18th 2010, 10:29 AM
ROOZ1234