Digital Convexity and Combinatorics on Words

TL;DR

This paper explores the combinatorial properties of digitally convex words, their geometric interpretations, and their connections to Christoffel and Sturmian words, focusing on operations like inflation and deflation.

Contribution

It formalizes the geometric properties of digitally convex words and investigates their relationships with well-known combinatorial word classes, introducing new insights into their structure.

Findings

Digital convex words relate closely to Christoffel and Sturmian words.

Operations of inflation and deflation are characterized for digitally convex words.

The study provides a combinatorial framework for understanding digital convexity in words.

Abstract

An upward (resp. downward) digitally convex word is a binary word that best approximates from below (resp. from above) an upward (resp. downward) convex curve in the plane. We study these words from the combinatorial point of view, formalizing their geometric properties and highlighting connections with Christoffel words and finite Sturmian words. In particular, we study from the combinatorial perspective the operations of inflation and deflation on digitally convex words.