Proving Convexity of S with f and X

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Discussion Overview

The discussion revolves around proving the convexity of the set S, defined as S={x from X: f(x)<=c}, where X is a convex set and f is a convex function. Participants explore the implications of these definitions and seek to clarify the proof of convexity, touching on theoretical aspects and definitions.

Discussion Character

  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • Some participants assert that S is convex based on the definitions of convex sets and functions.
  • One participant suggests that if a and b are in S, then the line segment between them must also be in S, leading to the conclusion that S is convex.
  • Another participant expresses uncertainty about the graphical representation of convex sets and functions, questioning what a convex subset of R looks like.
  • A later reply introduces a potential misunderstanding of the definition of a convex function, suggesting that the notion of convexity may differ among participants.
  • One participant attempts to clarify the proof by reiterating the conditions under which S is convex, referencing the properties of f and the relationship between points in S and X.

Areas of Agreement / Disagreement

Participants generally agree that S is convex based on the definitions provided, but there is some disagreement regarding the understanding of convex functions and their implications. The discussion remains somewhat unresolved as participants clarify their interpretations.

Contextual Notes

There are indications of differing interpretations of convex functions among participants, which may affect their understanding of the proof. Additionally, some participants express confusion about the graphical representation of convex sets.

Who May Find This Useful

Readers interested in the properties of convex sets and functions, as well as those studying mathematical proofs related to convexity in the context of real analysis.

jetoso
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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?
 
Last edited:
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jetoso said:
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 ...?
 
Sorry, the set S is convex.
 
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.
 
So how many convex subsets of R are there?
 
No, just prove that S is a convex set, given the definition of S.
 
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.
 
Last edited:
Must be like a ball or circle.
 
Check the edited post
 
  • #10
Well, sound like a midpoint of a linesegment
 
  • #11
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.
 
  • #12
Oh, I see. But well, how it looks like graphically? Is a line inside of a circle or something?
 
  • #13
is what line inside of what circle?
 
  • #14
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
 
Last edited:

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