Abstract

We define a new structure on a space endowed with convexities, and call it a fractoconvex structure (or, a space with fractoconvexity). We introduce two operations on a set of fractoconvexities and in a special case we show that they satisfy the laws for a distributive lattice. We establish a connection between fractoconvex sets and convex sets using the concept of independent convexities, based on the possibility of representing a fractoconvex set as the intersection of its convex hulls. Finally, we consider some examples of fractoconvexities on the 2-sphere and on set of integers.

Author information

Contact details are reproduced from the original publication and may be historical.

Aidar M. Dulliev

Kazan National Research Technical University, A. N. Tupolev University, Ulitsa Karla Marksa 10, Kazan 420111, Russia

dulliev@yandex.ru

Keywords

  • Convex sets
  • convex structures
  • convexity
  • abstract convexity
  • fractoconvexity
  • independent convexities

Mathematics Subject Classification

52A01, 52A30, 52A99

References

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Suggested citation

A. M. Dulliev. “Fractoconvex Structures.” Journal of Convex Analysis 26 (2019), No. 2, 527–536. https://doi.org/10.68381/jca26026

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Published by Heldermann Verlag, 2019. Rights now held by Banach Press LLC.