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

Printable View

- Feb 18th 2010, 08:52 AMROOZ1234Convex 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, 09:06 AMPlato
- Feb 18th 2010, 09:32 AMROOZ1234
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, 10:02 AMPlato
You certainly have all to components there.

I do not know the level of rigor required of you. - Feb 18th 2010, 11:29 AMROOZ1234
thx for your help