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Orthogonally connected sets

Xuemei He, Xiaotian Song, Liping Yuan, Tudor Zamfirescu

TL;DR

This work advances the theory of orthogonally and staircase connected sets by providing a comprehensive set of necessary and sufficient conditions for staircase connectedness in the plane, anchored in orthogonal convexity and local width criteria. It introduces a unimodal boundary framework that characterizes staircase connectedness via vertical cross-sections and boundary functions, and extends the discussion to convex bodies through the notion of obtuse geometry. The results unify geometric, combinatorial, and topological perspectives, with implications for related structures such as cylinders and convex hulls, and define the concept of s-extreme points to identify staircase endpoints. Overall, the paper enhances the toolkit for analyzing plane sets under orthogonal and staircase connectivity, with potential applications in digital geometry and related fields.

Abstract

In this paper, we further investigate the orthogonally connected sets and establish necessary and sufficient conditions for a set to be staircase connected.

Orthogonally connected sets

TL;DR

This work advances the theory of orthogonally and staircase connected sets by providing a comprehensive set of necessary and sufficient conditions for staircase connectedness in the plane, anchored in orthogonal convexity and local width criteria. It introduces a unimodal boundary framework that characterizes staircase connectedness via vertical cross-sections and boundary functions, and extends the discussion to convex bodies through the notion of obtuse geometry. The results unify geometric, combinatorial, and topological perspectives, with implications for related structures such as cylinders and convex hulls, and define the concept of s-extreme points to identify staircase endpoints. Overall, the paper enhances the toolkit for analyzing plane sets under orthogonal and staircase connectivity, with potential applications in digital geometry and related fields.

Abstract

In this paper, we further investigate the orthogonally connected sets and establish necessary and sufficient conditions for a set to be staircase connected.

Paper Structure

This paper contains 7 sections, 16 theorems, 2 equations, 11 figures.

Table of Contents

  1. Introduction
  2. Definitions and notation
  3. Orthogonally connected sets
  4. Staircase connectedness in the plane
  5. Staircase connectedness and convex bodies
  6. Characterization via unimodal functions
  7. $s$-extreme points

Key Result

Theorem 3.1

If $M\subset \mathds{R}^{2}$ is a convex set different from a strip, then $\complement M$ is orthogonally connected and $\mathrm{diam}_s(\complement M)\leq 4$.

Figures (11)

  • Figure 1:
  • Figure 2:
  • Figure 3: $b\in \mathrm{conv}([au\rangle \cup [av\rangle)$.
  • Figure 4: $a_1\in Y_b$.
  • Figure 5: $[a,a_1,a_2,a_3,b]$ is a staircase.
  • ...and 6 more figures

Theorems & Definitions (28)

  • Theorem 3.1
  • proof
  • Lemma 3.2
  • Theorem 3.3
  • proof
  • Theorem 3.4
  • proof
  • Theorem 3.5
  • Lemma 4.1
  • Theorem 4.2
  • ...and 18 more