Proving a convex function on an open convex set satisfies some inequalities

In summary, the problem asks to prove that for a convex function f defined on an open convex set O, the inequality f((1-t)a+tb)≤(1-t)f(a)+tf(b) holds for all a,b∈O and 0≤t≤1. The geometric interpretation of this result is straightforward. Attempts to use the mean value theorem in R^n have not been successful, suggesting a direct approach may be necessary. However, as this is typically given as a definition rather than a theorem, finding outside help has been challenging.
  • #1
michael.wes
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Homework Statement


Let [tex]f:\mathcal{O}\subset\mathbb{R}^n\rightarrow\mathbb{R}, \mathcal{O}[/tex] is an open convex set. Assume that [tex]D^2f(x)[/tex] is positive semi-definite [tex]\forall x\in\mathcal{O}[/tex]. Such [tex]f[/tex] are said to be convex functions.

Homework Equations


Prove that [tex]f((1-t)a+tb)\leq (1-t)f(a)+tf(b),a,b\in\mathcal{O},0 \leq t \leq 1[/tex] and interpret the result geometrically. (The interpretation is easy, it's the proof of the inequality I'm stuck with)

The Attempt at a Solution



In an earlier part of the question I proved that we have[tex]f(x)\geq f(a) + \grad f(a)\cdot (x-a) \forall x,a\in\mathcal{O}[/tex]

I have tried to use the mean value theorem in R^n in an attempt to link this with that result and the gradient, but it doesn't help since you lose information in using the mean value theorem, and this isn't an existence result, so it makes me think that there is a direct approach. This is typically given as the definition of a convex function on the web, and not a theorem, so I couldn't find help elsewhere.

Any help appreciated!

Edit: I'm still completely stuck.
 
Last edited:
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  • #2
Any help before tomorrow would be appreciated :)
 

1. What is a convex function?

A convex function is a type of function in mathematics where the line segment connecting any two points on the graph of the function lies above or on the graph. In simpler terms, it means that the function is always curving upwards and never has any downward curves or "dips".

2. What is an open convex set?

An open convex set is a type of set in mathematics where any two points within the set can be connected by a line segment that is completely contained within the set. This means that the set does not contain any "edges" or "corners" and is completely smooth.

3. How do you prove that a function is convex on an open convex set?

To prove that a function is convex on an open convex set, you must show that for any two points within the set, the line connecting them lies above or on the graph of the function. This can be done by using the definition of convexity and solving for the points in the function.

4. What are some common inequalities that a convex function on an open convex set satisfies?

Some common inequalities that a convex function on an open convex set satisfies include the Jensen's inequality, the Cauchy-Schwarz inequality, and the Hölder's inequality.

5. Why is proving convexity on an open convex set important?

Proving convexity on an open convex set is important because it allows us to better understand the behavior of a function and its properties. It also helps us to make predictions and draw conclusions about the function's behavior in various scenarios. Additionally, many real-world problems involve optimizing convex functions on open convex sets, so being able to prove convexity is crucial in solving these problems.

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