The Nearest Point Theorem for Weakly Convex Sets in Asymmetric Seminormed Spaces

  • Grigorii E. IvanovEmail author
  • Mariana S. Lopushanski
  • Maxim O. Golubev
Conference paper
Part of the Communications in Computer and Information Science book series (CCIS, volume 974)


Weakly convex sets in asymmetric seminormed spaces are considered. We prove that any point from some neighborhood of such a set has the unique nearest point in the set. The proof of the nearest point theorem is based on the theorem about the diameter of \(\varepsilon \)-projection which is also important in approximation theory. The notion of weakly convex sets in asymmetric seminormed spaces generalizes known notions of sets with positive reach, proximal smooth sets, and prox-regular sets. By taking the Minkowski functional of the epigraph of some convex function as a seminorm, the results obtained for weakly convex sets can be applied to weakly convex functions whose graphs are weakly convex sets with respect to this seminorm.


Weakly convex sets Asymmetric seminorm Metric projection 


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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Moscow Institute of Physics and Technology (State University)Dolgoprudny, Moscow RegionRussian Federation

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