TITLE: What are the main uses of convex functions?
QUESTION [26 upvotes]: 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.

REPLY [5 votes]: Convex analysis has proven that in a certain sense, convexity is the next best thing to linearity. Basically, as soon as you allow inequalities in linear setting, you are immediately in the territory of convex analysis. So convexity arises naturally in the study of dual spaces and weak topologies. In fact, there is a tendency to give the name "convexity" to anything that is nicely compatible with variational minimization problems. On another note, convexity and monotonicity are two sides of the same coin. Importance of monotonicity is something that does not need too much convincing. This link has been exploited e.g. in nonlinear PDEs, such as the Monge-Ampere equations.