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February 2016 The convex hull-like property and supported images of open sets
B. Ricceri
Ann. Funct. Anal. 7(1): 150-157 (February 2016). DOI: 10.1215/20088752-3428355


In this note, as a particular case of a more general result, we obtain the following theorem.

Let ΩRn be a nonempty bounded open set, and let f:Ω¯Rn be a continuous function which is C1 in Ω. Then, at least one of the following assertions holds:

(a) f(Ω)conv(f(Ω)).

(b) There exists a nonempty open set XΩ, with X¯Ω, satisfying the following property: for every continuous function g:ΩRn which is C1 in X, there exists λ˜0 such that, for each λ>λ˜, the Jacobian determinant of the function g+λf vanishes at some point of X.

As a consequence, if n=2 and h:ΩR is a nonnegative function, for each uC2(Ω)C1(Ω¯) satisfying in Ω the Monge–Ampère equation

uxxuyyuxy2=h, one has



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B. Ricceri. "The convex hull-like property and supported images of open sets." Ann. Funct. Anal. 7 (1) 150 - 157, February 2016.


Received: 3 April 2015; Accepted: 7 July 2015; Published: February 2016
First available in Project Euclid: 22 December 2015

zbMATH: 1336.35087
MathSciNet: MR3449347
Digital Object Identifier: 10.1215/20088752-3428355

Primary: 35B50
Secondary: 26A51‎ , 26B10 , 35F05 , 35F50 , 35J96

Keywords: convex hull property , Monge–Ampère equation , quasiconvex function , singular point , supported set

Rights: Copyright © 2016 Tusi Mathematical Research Group


Vol.7 • No. 1 • February 2016
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