Convex functions are real valued functions which visually can be understood as functions which satisfy the fact that the line segment joining any two points on the graph of the function lie above that of the function. Some familiar examples include , , etc.
These functions satisfy a number of interesting properties such as continuity,existence of left and right derivatives which are of usual interests to Analysis. These also lead to the notion of convexity being of interest in Optimization Theory, Probability Theory and the Calculus of Variations.
We first start with a preliminary definition of a convex set.
A set is said to be convex if given such that then
If then are all convex sets. Another interesting example is .
But, is not a convex set as but
We're now ready to define convex functions.
Let be a convex set and be a function.
Then is said to be convex if the following holds :
Note that is defined at as is convex. Also, after some routine computations, it is easy to see that this definition is equivalent to the intuitive definition given in the introduction.
Concave functions are defined as above but the inequality is flipped.
The most popular property of convex functions is the Jensen's Inequality.
Here's another way to check if a function is convex.
Let be a twice differentiable function.
- is convex on if and only if
- is concave on if and only if
This proof assumes the knowledge of the mean value theorem.
The proof for concave functions follows by the following result ;
If is concave, is convex.
Note that a function is convex if and only if the following holds :
The proof of this result is not difficult and hence is left to the interested reader.
By virtue of this result, we need to show that the above inequality holds.
To continue with the proof, put .
Now, consider the following expression :
This leads to the following:
And we're done.
Note: The condition that is non-negative is equivalent to the fact that is monotonically increasing. Hence, the theorem can only require the fact that is differentiable and is monotonically increasing.