About Convex Sets

dannyv · 2007-09-20 02:37:42

Hi, I'm new to this forum and I found it very interesting. So, as a first post I present this problem.

"Prove that the set X={(x,y) : ax+by <= c} is convex for any given a, b, and c."

How can I do that using only the axioms of vector spaces?

Thanks

John E. Franklin · 2007-09-20 02:43:35

This is doubly interesting because you have to prove this along the straight line, that is has no dents, and you have to prove there are no holes in the area out toward infinity, a weird subject.

dannyv · 2007-09-20 03:00:38

A convex set is a set X included in a vector space E where [a,b]={(1-t)a+tb : 0<=t<=1} and a,b in X implies [a,b] in X

dannyv · 2007-09-20 04:55:19

Here is a little proof:

JaneFairfax · 2007-09-20 08:23:38

dannyv wrote:

No! You cant subtract inequalities this way.

DOES NOT IMPLY

You can add inequalities normally, but you can subtract them like that.

Last edited by JaneFairfax (2007-09-20 08:25:17)

JaneFairfax · 2007-09-20 08:48:51

The way to complete the proof is as follows.

Adding [1] and [2] should give you what you are looking for.

dannyv · 2007-09-20 08:49:23

You are right!!

And what about this one:

Thanks a lot!!

dannyv · 2007-09-20 08:51:58

JaneFairfax wrote:

The way to complete the proof is as follows.
Adding [1] and [2] should give you what you are looking for.

Thanks:D

JaneFairfax · 2007-09-20 09:14:30

Youre welcome.

Math Is Fun Forum

#1 2007-09-20 02:37:42

About Convex Sets

#2 2007-09-20 02:43:35

Re: About Convex Sets

#3 2007-09-20 03:00:38

Re: About Convex Sets

#4 2007-09-20 04:55:19

Re: About Convex Sets

#5 2007-09-20 08:23:38

Re: About Convex Sets

#6 2007-09-20 08:48:51

Re: About Convex Sets

#7 2007-09-20 08:49:23

Re: About Convex Sets

#8 2007-09-20 08:51:58

Re: About Convex Sets

#9 2007-09-20 09:14:30

Re: About Convex Sets

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