Journal of Optimization Theory and Applications

, Volume 171, Issue 3, pp 865–886 | Cite as

(Convex) Level Sets Integration

  • Jean-Pierre CrouzeixEmail author
  • Andrew Eberhard
  • Daniel Ralph


The paper addresses the problem of recovering a pseudoconvex function from the normal cones to its level sets that we call the convex level sets integration problem. An important application is the revealed preference problem. Our main result can be described as integrating a maximally cyclically pseudoconvex multivalued map that sends vectors or “bundles” of a Euclidean space to convex sets in that space. That is, we are seeking a pseudoconvex (real) function such that the normal cone at each boundary point of each of its lower level sets contains the set value of the multivalued map at the same point. This raises the question of uniqueness of that function up to rescaling. Even after normalizing the function long an orienting direction, we give a counterexample to its uniqueness. We are, however, able to show uniqueness under a condition motivated by the classical theory of ordinary differential equations.


Convexity and pseudoconvexity Monotonicity and pseudomonotonicity Maximality Revealed preferences 

Mathematics Subject Classification

26B25 91B42 91B16 


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Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  • Jean-Pierre Crouzeix
    • 1
    Email author
  • Andrew Eberhard
    • 2
  • Daniel Ralph
    • 3
  1. 1.LIMOS, Campus Scientifique des CézeauxUniversité Blaise PascalAubièreFrance
  2. 2.School of Mathematical and Geospatial SciencesRMIT UniversityMelbourneAustralia
  3. 3.Judge Business SchoolUniversity of CambridgeCambridgeUK

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