Advertisement

Fenchel’s Unsolved Problem of Level Sets

  • Tamás Rapcsák
Chapter
  • 405 Downloads
Part of the Nonconvex Optimization and Its Applications book series (NOIA, volume 19)

Abstract

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.

Keywords

Utility Function Convex Function Positive Semidefinite Coordinate Representation Quasiconvex Function 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Copyright information

© Springer Science+Business Media Dordrecht 1997

Authors and Affiliations

  • Tamás Rapcsák
    • 1
  1. 1.Computer and Automation Institute of Hungarian Academy of SciencesBudapestHungary

Personalised recommendations