1. ## convex set

I'm self studying analysis and I'm going to need help from time to time conceptualizing the ideas. First off...Can someone touch on how they think about convex sets? If you could explain why all the points on a line between $x$ and $y$ is given by $\lambda{x}+(1-\lambda)y$ that would be helpful.

2. ## Re: convex set

Originally Posted by VonNemo19
I'm self studying analysis and I'm going to need help from time to time conceptualizing the ideas. First off...Can someone touch on how they think about convex sets?
You must be more exact in telling us exactly in what domain of definition you are working.
Are you working in a Euclidean metric, in a plane, is 3-space or general space?

Why that matters is clear from the description of convexity: in a convex region if $P~\&~Q$ are two points in the region and point $R$ is between them the point $R$ is also in the region. That is abstract in that it depends heavily upon the definition of betweenness.

In an ordinary Euclidean plane a set is convex if $P~\&~Q$ are two points in the set then the line segment $\overline{PQ}$ is a subset of the set. It is easy to see that a triangle with its interior is convex. The same for a circle, parallelogram, trapezoid. etc.

Think of a star shaped figure with its interior, it is easy to see that it is not convex. Pick two points in different peaks of the star. The line segment connecting them does not 'remain in' the star.

3. ## Re: convex setssetst

Given any two points in a convex set, every point on the line segment between them is in the set.

It is not true that all the points on a line between x and y are given by [tex]\lambda x+ (1- \lambda) y).

It is true that every point given by that is on the line passing through x and y but they are between x and y only for $0< \lambda< 1$. That it true because, first, this is linear in $\lambda$ so describes a straight line, second, when $\lambda= 0$ we get y, and, third, because when $\lambda= 1$ we get x. If $\lambda< 0$ we get points on the line but not "between" 0 and 1, they are on the side of y away from x. And if $\lambda> 1$ we get points on the line but on the side of x away from y.

4. ## Re: convex setssets

Thanks. You guys have helped enormously just talking about it with me for a second. The definition I'm working with is as follows:

We call a set $E\subset{R^k}$ convex if $\lambda\bold{x}+(1-\lambda)\bold{y}\in{E}$

whenever $\bold{x}\in{E}$ , $\bold{y}\in{E}$, and $0<\lambda<1$

I got the line idea from some website after I googled convex sets. I'm just trying to visualize what's happening.

5. ## Re: convex setssets

Originally Posted by VonNemo19
We call a set $E\subset{R^k}$ convex if $\lambda\bold{x}+(1-\lambda)\bold{y}\in{E}$ whenever $\bold{x}\in{E}$ , $\bold{y}\in{E}$, and $0<\lambda<1$
I'm just trying to visualize what's happening.
Allow me to be simply honest. You will never be completely comfortable with this material until you are comfortable working in general vector spaces. That is the point that Prof. HallsofIvey is making. $\bf{x}+\lambda(\bf{y}-\bf{x})$ where $\bf{x}~\&~\bf{y}$ are vectors and $\lambda$ is a scalar is a line in the space. But if $0\le\lambda\le 1$ is a line segment between $\bf{x}~\&~\bf{y}$. Please note the $\le$ is incorrect in your definition.

Try to visualize the difference in a line and a line segment. One is bounded and one is not bounded.
One is essential in the definition of convexity, the other does not.

6. ## Re: convex setssets setssets

Originally Posted by Plato
Allow me to be simply honest. You will never be completely comfortable with this material until you are comfortable working in general vector spaces. .
Honesty is what I'm after. Sugarcoating my inadequacy in certain areas will do nothing for me but slow me down. I need your replies to be matter of fact so that the space I work in is convex, allowing me to move in straight lines from points of ignorance to those of understanding. By the way, I skipped linear algebra because I found it dull. If I come up against something I can't get because of this, I'll take the time necessary to go back and learn it. I've done this with vectors already so that my understanding of convex sets is adequate for now.