Solarity of sets in max-approximation problems

  • Alexey R. AlimovEmail author


Let \({\mathrm {F}}_M\) be the max-projection operator (the farthest point mapping) associated with a given set M. A set M is called a local \(\max \)-sun at a point \(x_0\) if there is a farthest point \(\widehat{y}\in {\mathrm {F}}_Mx_0\) and a number \(\delta >0\) such that \( \widehat{y} \in {\mathrm {F}}_Mx\) for any point \(x\in \bigl [x_0, x_0+\delta (\widehat{y}-x_0)\bigr )\). A set M is called a local \(\max \)-sun on a set U if M is a local \(\max \)-sun at any point \(x_0\in U\). Some solarity and stability properties of the max-projection operator are established. Sufficient conditions in terms of the stability of the max-projection operator are given for a set to be a local \(\max \)-sun.


Farthest point max-approximation max-distance sun local max-sun 

Mathematics Subject Classification

Primary 41A65 Secondary 47H10 52A30 



The author is grateful to I. G. Tsar’kov for valuable comments and to the referee for noticing several misprints in the original manuscript and whose suggestions have saved other readers from a clumsy proof of Proposition 2.


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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Faculty of Mechanics and MathematicsMoscow State UniversityMoscowRussia
  2. 2.Steklov Mathematical InstituteRussian Academy of SciencesMoscowRussia

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