Convex function

File:ConvexFunction.svg

Convex function on an interval.

File:Epigraph convex.svg

A function (in black) is convex if and only if the region above its graph (in green) is a convex set.

In mathematics, a convex function is any function with value of the weighted average of 2 points is less than or equal to the weighted average of the function at those points. Also, a function is convex if and only if its epigraph is a convex set.

Examples

Three examples are:

• A line: [math]\displaystyle{ f(x) = x }[/math]
• A parabola: [math]\displaystyle{ f(x) = x^2 }[/math]
• Absolute value: [math]\displaystyle{ f(x) = |x| }[/math]