On Removable Sets For Convex Functions

Result code in IS VaVaI
<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F00216208%3A11320%2F14%3A10286397" target="_blank" >RIV/00216208:11320/14:10286397 - isvavai.cz</a>
Result on the web
<a href="http://dx.doi.org/10.1016/j.jmaa.2014.02.014" target="_blank" >http://dx.doi.org/10.1016/j.jmaa.2014.02.014</a>
DOI - Digital Object Identifier
<a href="http://dx.doi.org/10.1016/j.jmaa.2014.02.014" target="_blank" >10.1016/j.jmaa.2014.02.014</a>

Result language
angličtina
Original language name
On Removable Sets For Convex Functions
Original language description
In the present article we provide a sufficient condition for a closed set F in R^d to have the following property which we call c-removability: Whenever a function f:R^d->R is locally convex on the complement of F, it is convex on the whole R^d. We alsoprove that no generalized rectangle of positive Lebesgue measure in R^2 is c-removable. Our results also answer the following question asked in an article by Jacek Tabor and Jozef Tabor [J. Math. Anal. Appl. 365 (2010)]: Assume the closed set F in R^d issuch that any locally convex function defined on R^dF has a unique convex extension on R^d. 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 R^2.
Czech name
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Czech description
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Type
J<sub>x</sub> - Unclassified - Peer-reviewed scientific article (Jimp, Jsc and Jost)
CEP classification
BA - General mathematics
OECD FORD branch
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Project
<a href="/en/project/GCP201%2F10%2FJ039" target="_blank" >GCP201/10/J039: Curvature measures and integral geometry</a><br>
Continuities
I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace

Publication year
2014
Confidentiality
S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

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