Proving Convexity of S with f and X

[jetoso]

Given a convex set X and a convex function f: X - R, show that for any c from R, the set S={x from X: f(x)<=c} is convex

Any advice about how to prove it?

 

[Timbuqtu]

Given a convex set X and a convex function f: X - R, show that for any c from R, the set S={x from X: f(x)<=c}

The set S is ...?

 

[jetoso]

Sorry, the set S is convex.

 

[jetoso]

Convexity

The set X is convex if for any x, y from X, we have that the line segment joining x and y: ax + (1-a)y, also belongs to X, for any scalar a from (0, d], d > 0.

 

[matt grime]

So how many convex subsets of R are there?

 

[jetoso]

No, just prove that S is a convex set, given the definition of S.

 

[matt grime]

Erm, yeah, but it appears obvious. If a and b are in S, then the line segment between them is in X, hence the image of the line segment is a convex subset of R, a and b both satisfy f(a) and f(b) <=c so, I repeat, what does a convex subset of R look like?

EDIT think i have a different notion of a convex function than you. I'm guessing you mean that f is convex if for each a and b and x any point on the line segment a to b then f(x) < = (f(a)+f(b))/2, but that makes it even easier.

 

[jetoso]

Must be like a ball or circle.

 

[matt grime]

Check the edited post

 

[jetoso]

Well, sound like a midpoint of a linesegment

 

[matt grime]

what sounds like a midpoint of what linesegment?

if x is on the line segment from a to be and a and b are in S, then f(x) <= (f(a)+f(b))/2 <= (c+c)/2 = c hence x is S. Thus S is convex.

 

[jetoso]

Oh, I see. But well, how it looks like graphically? Is a line inside of a circle or something?

 

[matt grime]

is what line inside of what circle?

 

[jetoso]

I think I am losing the point here, sorry about that. So, the point here is that, if S is a convex subset of R, and X is a convex subset of R, and for both of them exists a function f, then for any two points x and y from X, they also belong to S such that S = {x from X: f(x)<=c}.
In such a way that:
f(ax+(1-a)y)<=af(x)+(1-a)f(y)
then
<=ac+(1-a)c = c
for which f(x)<=c, this implies that S is convex.
Right? Sorry if I am wasting you time... =S