Fractoconvex structures

TL;DR

This paper introduces fractoconvex structures on spaces with convexities, explores their algebraic properties, and connects them to convex sets through independent convexities, with examples on spheres and integers.

Contribution

It defines fractoconvex structures, introduces operations forming a distributive lattice, and links fractoconvex sets to convex sets via independent convexities.

Findings

Fractoconvex structures form a distributive lattice under certain operations.

Fractoconvex sets can be represented as intersections of convex hulls.

Examples include fractoconvexities on the 2-sphere and on the integers.

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 $\mathbb Z$.