Blaschke and Separation Theorems for Orthogonally Convex Sets

TL;DR

This paper explores the geometric and topological properties of orthogonally convex sets, establishing new theorems for their separation and representation in the plane and higher dimensions.

Contribution

It introduces a Blaschke-type theorem and a separation method for orthogonally convex sets, along with their representation via staircase-halfplanes.

Findings

Established a Blaschke-type theorem for orthogonally convex sets in the plane.

Proved that orthogonally convex sets can be represented as intersections of staircase-halfplanes.

Analyzed topological properties of orthogonally convex sets in higher dimensions.

Abstract

In this paper, we deal with analytic and geometric properties of orthogonally convex sets. We establish a Blaschke-type theorem for path-connected and orthogonally convex sets in the plane using orthogonally convex paths. The separation of these sets is established using suitable grids. Consequently, a closed and orthogonally convex set is represented by the intersection of staircase-halfplanes in the plane. Some topological properties of orthogonally convex sets in dimensional spaces are also given.