Help with convex function properties

[pamparana]

Hello all,

I am reading a paper and there is one bit in the paper that I am having a bit of trouble understanding.

Say V(a) is a convex function and then the paper has the following line:

[V(-2pi + a] + V(2pi + a]] >= 2V(a)

I am sure this relationship is simple and falls out somehow from the definition of convex functions but I am unable to convince myself of it.

I would be very grateful if someone can help me with figuring out why this relationship holds.

Many thanks,

Luca

 

[viraltux]

Hello all,

I am reading a paper and there is one bit in the paper that I am having a bit of trouble understanding.

Say V(a) is a convex function and then the paper has the following line:

[V(-2pi + a] + V(2pi + a]] >= 2V(a)

I am sure this relationship is simple and falls out somehow from the definition of convex functions but I am unable to convince myself of it.

I would be very grateful if someone can help me with figuring out why this relationship holds.

Many thanks,

Luca

Hi Luca,

Well, if given two points in a convex function you can always find another point between these two with a lower or equal value and you have V(-2pi + a) + V(2pi + a) >= 2V(a) it simply tells you that a is one of these points between -2pi+a and 2pi+a... right?

 

[pamparana]

Of course. I am so stupid!

Thanks!
Luca

 

[viraltux]

Of course. I am so stupid!

Thanks!
Luca

 

[blue_raver22]

Hello Luca,

I can understand your confusion with this relationship. Let me try to explain it to you in simple terms.

First, let's define what a convex function is. A convex function is a function where the line segment connecting any two points on the graph of the function always lies above or on the graph. In other words, the function is always "curving upwards" and never "curving downwards."

Now, let's look at the relationship in the paper: [V(-2pi + a] + V(2pi + a]] >= 2V(a). This means that the sum of the values of the function at -2pi + a and 2pi + a is greater than or equal to twice the value of the function at a.

To understand why this relationship holds, let's look at a graph of a convex function. The graph of a convex function will always have a "bowl" shape, with the lowest point at the bottom. Now, if we take any three points on this graph (a, -2pi + a, and 2pi + a), we can see that the line segment connecting -2pi + a and 2pi + a will always lie above or on the graph. This is because the function is convex, and the line segment connecting any two points on the graph will always lie above or on the graph.

Since the line segment connecting -2pi + a and 2pi + a lies above or on the graph, the sum of the values of the function at these two points will be greater than or equal to the value of the function at a. This is because the graph is "curving upwards," and the sum of the values at the two points will always be greater than or equal to the value at the lowest point (a).

I hope this explanation helps you understand why this relationship holds. If you have any further questions, please let me know.

Best,