On the Continuous Representation of Quasiconcave Functions by Their Upper Level Sets

Philippe Bich

doi:10.68381/jca17008

Abstract

We provide a continuous representation of quasiconcave functions by their upper level sets. A possible motivation is the extension to quasiconcave functions of a result by D. H. Hyers and S. M. Ulam [Proc. Amer. Math. Soc. 3(5) (1952) 821–828], which states that every approximately convex function can be approximated by a convex function.

Author information

Contact details are reproduced from the original publication and may be historical.

Philippe Bich

Paris School of Economics, Centre d'Economie de la Sorbonne, Université Paris I Panthéon, 106/112 Boulevard de l'Hôpital, 75013 Paris, France

philippe.bich@univ-paris1.fr

Keywords

Quasiconcave
upper level set

References

D. H. Hyers, S. M. Ulam: Approximately convex functions, Proc. Am. Math. Soc. 3 (1952) 821–828.https://doi.org/10.1090/S0002-9939-1952-0049962-5 Google Scholar
J.-P. Crouzeix: Continuity and differentiability of quasiconvex functions, Chapter 3 in: Handbook of Generalized Convexity and Generalized Monotonicity, N. Hadjisavvas et al. (ed.), Nonconvex Optimization and its Applications 76, Springer, New York (2005) 121–149.https://doi.org/10.1007/0-387-23393-8_3 Google Scholar

Suggested citation

Published by Heldermann Verlag, 2010. Rights now held by Banach Press.

[1] D. H. Hyers, S. M. Ulam: Approximately convex functions, Proc. Am. Math. Soc. 3 (1952) 821–828.https://doi.org/10.1090/S0002-9939-1952-0049962-5 Google Scholar

[2] J.-P. Crouzeix: Continuity and differentiability of quasiconvex functions, Chapter 3 in: Handbook of Generalized Convexity and Generalized Monotonicity, N. Hadjisavvas et al. (ed.), Nonconvex Optimization and its Applications 76, Springer, New York (2005) 121–149.https://doi.org/10.1007/0-387-23393-8_3 Google Scholar