What are the main uses of Convex Functions?

Question

Up till now I have just learned that the concept of convexity in functions of one variable is used to complete the graphs of functions, meaning to locate points of inflexion and see if the graph is concave or convex in given intervals. Spivak's book mention that altough this is what we typically learn at calculus, the importance of this concept doesn't lie precisely on ploting graphs and it is really worth to assimilate the information. I'd like to know the main aplications of this concept and it's importance in general.

Answer 1

In short, convex functions are easy to work with even when not C^2. For example, many theorems about L_p spaces often abuse the convexity property.

Answer 2

The concept is used in Economic Theory.