Convex sets - How do we get (1−t)x+ty

[bhagwad]

In definition 2.17 of Rudin's text, he says that a set E is convex if for any two points x and y belonging to E, (1−t)x+ty belongs to E when 0<t<1.

I learned that this means the point is between x and y. But I'm not able to see this intuitively. Can anyone help me "see" this?

 

[spamiam]

How would you parametrize the line segment between the points $x$ and $y$? If you know how to do that, I think you'll see the answer.

 

[economicsnerd]

Draw two vectors x,y on a copy of \mathbb R^2. Or on a piece of paper, if that's all you have available.

Draw an arrow from x to y on your paper. Its tail should be at x and its head at y. We can think of the arrow as the vector z:= y-x. Indeed, if we moved the arrow so that its tail is at the origin, the head would lie at z.

-Imagine an ant is sitting at x and you want it to travel to y in a straight line. What should it do? It should travel all the way along z. This would bring it to x + z, also known as y.
-Now, imagine the ant is sitting at x again, and you want it to travel \frac13 of the way to y in a straight line. What should it do now? It should travel \frac13 of the way along z. This would bring it to x + \frac13z.
-And what if you want it to travel (again, from x) \frac57 of the way to y in a straight line? Now, it should go to x + \frac57z

This is the sense in which \{x+tz: t\in[0,1]\} parametrizes the line segment between x and y. Finally, notice that x+tz = (1-t)x + ty.

 

[bhagwad]

Draw two vectors x,y on a copy of \mathbb R^2. Or on a piece of paper, if that's all you have available.

Draw an arrow from x to y on your paper. Its tail should be at x and its head at y. We can think of the arrow as the vector z:= y-x. Indeed, if we moved the arrow so that its tail is at the origin, the head would lie at z.

-Imagine an ant is sitting at x and you want it to travel to y in a straight line. What should it do? It should travel all the way along z. This would bring it to x + z, also known as y.
-Now, imagine the ant is sitting at x again, and you want it to travel \frac13 of the way to y in a straight line. What should it do now? It should travel \frac13 of the way along z. This would bring it to x + \frac13z.
-And what if you want it to travel (again, from x) \frac57 of the way to y in a straight line? Now, it should go to x + \frac57z

This is the sense in which \{x+tz: t\in[0,1]\} parametrizes the line segment between x and y. Finally, notice that x+tz = (1-t)x + ty.

Thank you very much! I actually drew the lines on the paper and am trying to figure it out. Excellent explanation!