Convexity and Concavity

This article will explore convexity and concavity, both in functions and polygons. When we talk about convexity and concavity, we are referring to the shape of the curve or function. Now, convexity and concavity do sound like complicated terms. However, we will learn that they are nothing so fearful- they are simply a way of describing what a curve or function looks like. So, without any further introduction, let's define what exactly concave and convex functions are.

What is a Concave Function?

First, we will discuss concave functions.

Shown below is an example of a concave function. You can see that if we pick any two points on the curve, and draw a line segment between them, the line segment will always lie below the curve.

Example of a concave function- StudySmarter Originals

How to find concavity in an equation

To algebraically portray that a function is concave, the following equation is used:

$\lambda x+(1-\lambda )y⩽\lambda f\left(x\right)+(1-\lambda )f\left(y\right)$

Before breaking this down into words, it is important to understand that$\lambda x+(1-\lambda )y$selects any point between the points$x$and$y$. Similarly, $\lambda f\left(x\right)+(1-\lambda )f\left(y\right)$ selects any point between$f\left(x\right)$and$f\left(y\right)$.

The first portion of the inequality finds the value of the function of any point between$x$and$y$. The second portion selects any point between the functions of $x$ and $y$. Thus, what this equation represents is that the function of any point between$x$and$y$is greater than or equal to any point between points$x$and$y$.

Observing the graph above, it is clear that this is true for that graph as the functions of points between the coordinates of intersection are above the functions between both functions, represented by the blue linear equation.

What is a Convex Function?

An example can be seen below:

Example of convex function- StudySmarter Originals

It is clear that this type of function opposes a concave function. A line between two points (representing all functions between both functions) is always above or at the same level as the function itself.

How to find Convexity in an Equation

Since both types of functions have similar parameters, their equations resemble each other. They have only one vital difference:

Notice the inequality sign is flipped the other way. Since all other components are identical, this function represents that any point between the selected coordinates is greater than or equal to any point on the function between both coordinates.

Can a Function be both Concave and Convex?

Yes, this is possible. This is because both functions have an equal sign in the inequality. The most common example of this is any straight line, as the function for a point between any two points will match the equivalent function between both functions.

What are Concave Polygons?

An example of this type of shape can be seen below:

Example of concave polygon- StudySmarter Originals

In the above shape, the angle EDC exceeds 180 degrees. Therefore, it is a concave function.

There is a visual test that can be done to check for a concave polygon: if a straight line between any two points inside a polygon goes outside of the shape, that shape is a concave polygon. For example, below we can see that if we draw in the straight line segment $EC$, the line goes outside of the shape. Therefore, the polygon is concave.

Visual test for concave polygon- StudySmarter Originals

What are Convex Polygons?

An example would be the following shape:

Convex polygon example- StudySmarter Originals

The visual test for concave polygons can be inverted to test for a convex polygon. Since this polygon has no two points which create a segment that crosses outside of it, this geometric shape is a convex polygon.

Differences between concavity and convexity

The main difference between concavity and convexity is the fact that the angles subtended in convex shapes curve outwards whereas the angles subtended in concave shapes curve inwards. This is all based on whether or not there is an angle that exceeds 180 degrees.

Below are some further examples of concave and convex polygons. See if you can determine whether they are concave or convex.

Below are some further examples of concave and convex functions. See if you can determine whether they are concave or convex.