A Note on Quasi-Convex Functions

Abstract

We present an example of smooth quasi-convex functions in the positive octant of

\mathbb{R}^3

which cannot be obtained as the images of convex smooth functions under a monotone smooth mapping of

\mathbb{R}

.

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

127 Vincent Hall, University of Minnesota, Minneapolis, MN 55455, U.S.A.

26B25, 26B40

S. Boyd, L. Vandenberghe: Convex Optimization, Cambridge University Press, Cambridge (2009).Google Scholar
B. de Finetti: Sulle stratificazioni convesse, Ann. Mat. Pura Appl. (4), Vol. 30 (1949) 173–183.https://doi.org/10.1007/BF02415006 Google Scholar
H. J. Greenberg, W. P. Pierskalla: A review of quasi-convex functions, Operations Research 19(7) (1971) 1553–1570.https://doi.org/10.1287/opre.19.7.1553 Google Scholar
N. V. Krylov: On the general notion of fully nonlinear second order elliptic equation, Trans. Amer. Math. Soc. 347(3) (1995) 857–895.https://doi.org/10.1090/S0002-9947-1995-1284912-8 Google Scholar
P. G. Moffatt: A class of indirect utility functions predicting Giffen behavior. New insights into the theory of Giffen goods, in: Lecture Notes Econ. Math. Systems 655, Springer, Berlin et al. (2012) 127–141.https://doi.org/10.1007/978-3-642-21777-7_10 Google Scholar
A. W. Roberts, D. E. Varberg: Convex Functions, Pure and Applied Mathematics 57, Academic Press, New York et al. (1973).Google Scholar

Published by Heldermann Verlag, 2019. Rights now held by Banach Press LLC.