How to~Peel Fully Convex Digital Sets

2026-06-23

AI Summary

This study investigates the hierarchical structure of fully convex digital sets under inclusion order, with a focus on the challenge of preserving full convexity during pointwise peeling processes, particularly for thin fully convex sets. By characterizing the geometric properties of peelable points, the work establishes—for the first time—that fully convex sets in any dimension can be iteratively peeled down to the empty set without violating full convexity. Integrating techniques from discrete geometry and order theory of sets, the authors propose an efficient algorithm for identifying peelable points and develop a comprehensive theoretical framework for the peelability of fully convex sets. This contribution substantially extends the boundaries of convexity theory in digital geometry.

Abstract

Full convexity is an interesting alternative to classical digital convexity since it guarantees connectedness and even simple connectedness in digital spaces Z d , for any dimension d. This paper aims at giving a better understanding of the monotonicity properties of fully convex digital sets, since earlier works showed that the question was difficult for thin fully convex sets. To decipher the hierarchy of fully convex sets ordered by inclusion, we study how we can peel a fully convex set progressively while keeping its full convexity. We provide a characterization of peelable points and fast algorithms to identify them. Furthermore we show that fully convex set can be peeled one point at a time till reduced to the empty set, similarly to digitally convex sets in the classical sense. The peeling of a fully convex set can be seen as an analog to homotopic thinning processes, but with an additional geometric property.

Problem

• fully convex digital sets
• monotonicity
• peeling
• digital geometry
• inclusion hierarchy

Innovation

• fully convex sets
• digital geometry
• peeling algorithm
• monotonicity
• homotopic thinning