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Convex: Definition, Shape & Function

Instructor: Laura Pennington

Laura has taught collegiate mathematics and holds a master's degree in pure mathematics.

In this lesson, we will discuss the definition of convex, look at convex shapes and convex functions, and learn how to identify them through rules and examples.

Convex Definition

Take a look at this pizza. Notice the pizza is a circle, and it is curved outward everywhere: this pizza is convex. When something is said to be convex, it means that it is curved outwards.

Convex Shape
convex1-5

However, if we were to take a slice of pizza out, then the remaining pizza would no longer be convex. It would be curved inward where the slice was taken out.

Concave Shape
convex1-6

When a shape or object is curved inwards like the pizza with the slice removed, we say the object is concave. In this lesson, we will concentrate on convex, and look at what it means for shapes and functions to be convex.

Convex Shapes and Polygons

To be convex is to be curved outwards. This definition makes it fairly clear what it means for a shape to be convex. It must be curved outwards everywhere, just like our pizza in the first example. Formally, in order for a shape to be convex, we must be able to draw two points anywhere within the shape, and the line connecting them cannot go outside of the shape. If we can connect two points within a shape where the line connecting them goes outside the shape, then the shape is not convex.

Convex and Concave Shapes
convex 1-3

When a shape is a polygon, there is another rule for determining if it is convex. A polygon is a two-dimensional shape with straight lines as sides. A polygon is convex if all of its internal angles are less than or equal to 180 degrees.

Convex and Concave Polygons
convex4

Convex Functions

Now that we're familiar with convex shapes, let's consider convex functions. In mathematics, functions can be classified as convex based on how their slopes behave. As a review, the slope of a function is how quickly the function is increasing or decreasing: it is the rate of change of y with respect to x.

A function is said to be convex when the function has a slope that is increasing. For example, consider the function y = x^2.

Convex Function
convex5

Notice that the slope of the function y = x^2 is increasing as x increases. Therefore, the function is convex.

Another way to recognize this is by the concavity of the function. The concavity of a function also has to do with its slope. When the slope of a function is increasing, we say that the function is concave up. When the slope of a function is decreasing, we say that the function is concave down. Notice that the definition of concave up and convex are the same. Therefore, when a function is convex, we can also say that it is concave up.

Examples

1.) Is a square convex? Explain.

convex8

Solution:

A square is convex! This can be shown in two ways. Notice that we are unable to connect any two points within the square and have the line connecting them go outside the shape. Since a square is a polygon, we should also note that all of the square's interior angles are 90 degrees, and thus all less than 180 degrees.

convex9

Therefore, a square is convex.

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