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Journal of Convex Analysis 26 (2019), No. 2, 527--536
Copyright Heldermann Verlag 2019

Fractoconvex Structures

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

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.

Keywords: Convex sets, convex structures, convexity, abstract convexity, fractoconvexity, independent convexities.

MSC: 52A01, 52A30, 52A99

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