Thread: Question about convex set

So i know the definition of a convex set:
Set S is convex then tx+(1-t)y belongs to S for all 0<t<1

I have to decide whether the following set is convex:
{(x,y); x-y<=1}

I started with tx+(1-t)y = t(x-y)+y <= t+y

then I am stuck.

edit: hm I think I might have been mixing up the the points that I choose (x,y) with the variabels in x-y<=1...

I redid it with matrices and this is what i came up with:

(1 -1)(x1,x2) <= 1 , c = (1 -1)
cx <= 1
cy <= 1

=> c(tx + (1-t)y) = tcx+(1-t)cy <= t + 1 -t = 1

Is this the correct way to solve the problem?

Originally Posted by MagisterMan

So i know the definition of a convex set:
Set S is convex then tx+(1-t)y belongs to S for all 0<t<1

I have to decide whether the following set is convex:
{(x,y); x-y<=1}

I started with tx+(1-t)y = t(x-y)+y <= t+y

then I am stuck.

edit: hm I think I might have been mixing up the the points that I choose (x,y) with the variables in x-y<=1... Yes, that is exactly what you were doing!

In this problem, the set is a set of points in two-dimensional space. The definition of convexity says that you have to take two points in S. Call them and . The fact that these points are in S tells you that and

To see whether S is convex, you have to check whether the point



is in S. The condition for that is So you need to check whether that condition follows from what you are given.

Super helpful! Thanks