Fenchel’s Unsolved Problem of Level Sets
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It is well-known that a convex function has convex less-equal level sets. That the converse is not true was realized by de Finetti (1949). The problem of level sets, discussed first by Fenchel in 1953, is as follows: Under what conditions is the family of level sets of a convex function a nested family of closed convex sets? Fenchel (1953, 1956) gave necessary and sufficient conditions for the existence of a convex function with the prescribed level sets and the existence of a smooth convex function under the assumption that the given subsets are the level sets of a twice differentiable function. In the first case, seven conditions were deduced, and while the first six are simple and intuitive, the seventh is rather complicated. This fact and the additional assumption in the smooth case, according to which the given subsets are the level sets of a twice differentiable function, seem to be the motivation that Roberts and Varberg (1973, p. 271) drew up anew the following problem of level sets: “What “nice” conditions on a nested family of convex sets will ensure that it is the family of level sets of a convex function?” In the sequel, the notions of convexifiability and concavifiability are used as synonyms, because if a function f is convex, then –f concave.
KeywordsUtility Function Convex Function Positive Semidefinite Coordinate Representation Quasiconvex Function
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