On removable sets for convex functions |
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Authors: | Dušan Pokorný Martin Rmoutil |
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Institution: | Charles University, Faculty of Mathematics and Physics, Sokolovská 83, 186 75 Praha 8 Karlín, Czech Republic |
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Abstract: | In the present article we provide a sufficient condition for a closed set F∈Rd to have the following property which we call c -removability: Whenever a continuous function f:Rd→R is locally convex on the complement of F , it is convex on the whole Rd. We also prove that no generalized rectangle of positive Lebesgue measure in R2 is c-removable. Our results also answer the following question asked in an article by Jacek Tabor and Józef Tabor (2010) 5]: Assume the closed set F⊂Rd is such that any locally convex function defined on Rd?F has a unique convex extension on Rd. Is F necessarily intervally thin (a notion of smallness of sets defined by their “essential transparency” in every direction)? We prove the answer is negative by finding a counterexample in R2. |
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Keywords: | Convex function Locally convex function Intervally thin set c-removable set Convex extension Separately convex function |
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