Concave utility functions are widely used in applied economics and in economic theory. Their importance stems to a large extent from their level set being convex. De Finetti found that the converse is not true, and Fenchel investigated the matter further. We present a complete solution to this problem. Actually there are three approaches, depending on whether you look at values of the functions, their gradients, or their Hessian. We construct least concave functions and discuss their economic significance.
