# Convex Functions: Info on Minima, Reconstructing Original Functions

• MHB
• moyo
In summary, convoluting two convex functions results in a convex function and the positions of the minima can provide information about the original functions, but it may not be possible to fully reconstruct them from just the graph of their composition.
moyo
If you would allow me to ask...

if i have two convex functions , and i was to place one inside the other, i.e. convolute them...what could be said in general about the resultant function.

what information about the original functions can be taken from the positions of the minima.

and is there a way to reconstruct the original functions if given only the graph of their composition.

Thankyou

for your question! The resulting function from convoluting two convex functions will also be a convex function. This is because the sum of two convex functions is also convex.

The positions of the minima can provide information about the structure of the original functions. For example, if the minima of the resulting function occur at the same positions as the minima of one of the original functions, it could indicate that the other original function is relatively flat in that region.

However, without knowing the specific equations of the original functions, it may not be possible to fully reconstruct them from just the graph of their composition. This is because there could be multiple combinations of functions that could result in the same graph when convoluted.

In order to fully reconstruct the original functions, you would need more information such as the specific equations or additional data points. I hope this helps answer your questions!

## What are Convex Functions?

Convex functions are mathematical functions that have a specific shape and properties. They are defined as functions where the line segment connecting any two points on the graph of the function lies above or on the graph itself. In other words, the function is always "curving up" and does not have any local maxima or minima points.

## What is the importance of Convex Functions?

Convex functions have many applications in mathematics, economics, and engineering. They are used to model and solve optimization problems, such as finding the minimum or maximum of a function. They also have important properties that make them easier to work with and analyze compared to non-convex functions.

## What is the difference between a global minimum and a local minimum?

A global minimum is the lowest point on the entire graph of a function, while a local minimum is the lowest point in a specific region of the graph. In convex functions, the global minimum is also the local minimum, meaning it is the lowest point on the entire graph and in every region of the graph.

## How can convex functions be used to reconstruct original functions?

Convex functions can be used to approximate and reconstruct original functions. This is done by finding a convex function that closely follows the original function in a given interval. The closer the convex function is to the original function, the better the reconstruction.

## Can convex functions have multiple minima?

No, convex functions only have one global minimum. This is because the function is always "curving up" and does not have any local minima points. Therefore, there is only one point on the graph that is the lowest point, making it the global minimum.

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