Multi-convex sets in real projective spaces and their duality

TL;DR

This paper explores the structure and duality of multi-convex sets in real projective spaces, revealing a duality that can simplify geometric problems in computational geometry.

Contribution

It introduces a duality theory for saturated multi-convex sets in real projective spaces, linking them via order anti-isomorphism to dual space counterparts.

Findings

Established a duality between multi-convex sets in projective spaces and their duals.

Showed that intersections of convex sets can split into multiple convex components.

Proposed potential applications in transforming geometric problems for easier solutions.

Abstract

We study intersections of projective convex sets in the sense of Steinitz. In a projective space, an intersection of a nonempty family of convex sets splits into multiple connected components each of which is a convex set. Hence, such an intersection is called a multi-convex set. We derive a duality for saturated multi-convex sets: there exists an order anti-isomorphism between nonempty saturated multi-convex sets in a real projective space and those in the dual projective space. In discrete geometry and computational geometry, these results allow to transform a given problem into a dual problem which sometimes is easier to solve. This will be pursued in a later paper.