Abstract
A quasiconvex function f being given, does there exist an increasing and continuous function k which makes \(k\circ f\) 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.
Highlights
A function f : Rn → R is said to be quasiconvex when its level sets St(f ) = { x : f (x) ≤ t} are convex
This utility function u is unique up to a scaling function k: if u is a utility function describing the behavior of the consumer, so is k ◦ u
Given a quasiconvex function f, we study what conditions on k are necessary to get k ◦ f convex, least convex
Summary
A function f : Rn → R is said to be quasiconvex when its (lower) level sets St(f ) = { x : f (x) ≤ t} are convex. B) Let f be a proper lsc convex function on Rn. Set C = epi(f ). The function f ∗ defined by f ∗(x∗) = δC∗ (x∗, −1) is lsc, proper and convex. C) let f be a lsc quasiconvex function on Rn. By definition, its level sets St(f ) = { x : f (x) ≤ t}, t ∈ R are closed and convex. An improvement of the second statement, due to Fenchel, says that the nonempty lower level sets of a proper lsc convex function have the same barrier cones.
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