In the context of the financial value ascribed to the ability to hold off on deciding whether to make an investment, Jensen's inequality has been used to describe the fact that the average of the maximum returns from several possible future outcomes is greater than the maximum of the average returns for those same set of possible future outcomes. This is because (apparently) the max function is convex. However, I've only seen Jensen's inequality described in terms of a function that takes one input argument and returns one output argument i.e. a transformation of the outcomes values of a random variable. The max function is not like that. So it's not possible to plot the max function as output value versus input value. For this reason, the justification for calling the max function convex eludes me. In what sense is the max function convex?