| # Proving convexity via monotonicity Consider the function , with values . We can express the function as the composition of the function with the function with values That is, . Since belongs to the domain of for every , the domain of is indeed the whole real line. The functions are both convex, and is monotone increasing (note that the domain of is chosen to ensure monotonicity). Hence, by the monotonicity property, the composition is also convex.
| The picture shows the set (in three dimensions), and its projection on the space of -variables, which is the epigraph of , . The epigraph of is the projection of the same set on the space of -variables.