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On Quasiconvex Functions Which are Convexifiable or Not

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  • Jean-Pierre Crouzeix

    (LIMOS, Université Clermont Auvergne)

Abstract

A quasiconvex function f being given, does there exist an increasing and continuous function k which makes $$k\circ f$$ k ∘ f convex? How to build such a k? Some words on least convex (concave) functions. The ratio of two positive numbers is neither locally convexifiable nor locally concavifiable. Finally, some considerations on the approximation of a preorder from a finite number of observations and on the revealed preference problem are discussed.

Suggested Citation

  • Jean-Pierre Crouzeix, 2022. "On Quasiconvex Functions Which are Convexifiable or Not," Journal of Optimization Theory and Applications, Springer, vol. 193(1), pages 66-80, June.
    • Handle: RePEc:spr:joptap:v:193:y:2022:i:1:d:10.1007_s10957-021-01965-1
    DOI: 10.1007/s10957-021-01965-1
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    References listed on IDEAS

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    1. J. P. Crouzeix, 1980. "Conditions for Convexity of Quasiconvex Functions," Mathematics of Operations Research, INFORMS, vol. 5(1), pages 120-125, February.
    2. Bolte, Jérôme & Pauwels, Edouard, 2020. "Curiosities and counterexamples in smooth convex optimization," TSE Working Papers 20-1080, Toulouse School of Economics (TSE).
    3. Debreu, Gerard, 1976. "Least concave utility functions," Journal of Mathematical Economics, Elsevier, vol. 3(2), pages 121-129, July.
    4. W. E. Diewert, 1973. "Afriat and Revealed Preference Theory," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 40(3), pages 419-425.
    5. Jean-Pierre Crouzeix & Andrew Eberhard & Daniel Ralph, 2016. "(Convex) Level Sets Integration," Journal of Optimization Theory and Applications, Springer, vol. 171(3), pages 865-886, December.
    6. Newman, Peter, 1969. "Some properties of concave functions," Journal of Economic Theory, Elsevier, vol. 1(3), pages 291-314, October.
    7. Kannai, Yakar, 1974. "Approximation of convex preferences," Journal of Mathematical Economics, Elsevier, vol. 1(2), pages 101-106, August.
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