On k-Convex Polygons

TL;DR

This paper introduces the concept of k-convex polygons, provides recognition algorithms for 2-convex polygons, characterizes their shape, and explores the combinatorial complexity of their geometric permutations.

Contribution

It defines k-convexity in polygons, offers recognition algorithms for 2-convex polygons, and analyzes their shape and permutation complexity.

Findings

Recognition of 2-convex polygons can be done in O(n log n) time.

2-convex polygons can be characterized and described geometrically.

The number of generalized geometric permutations can be exponential in the number of 2-convex objects.

Abstract

We introduce a notion of $k$-convexity and explore polygons in the plane that have this property. Polygons which are \mbox{$k$-convex} can be triangulated with fast yet simple algorithms. However, recognizing them in general is a 3SUM-hard problem. We give a characterization of \mbox{$2$-convex} polygons, a particularly interesting class, and show how to recognize them in \mbox{$O(n \log n)$} time. A description of their shape is given as well, which leads to Erd\H{o}s-Szekeres type results regarding subconfigurations of their vertex sets. Finally, we introduce the concept of generalized geometric permutations, and show that their number can be exponential in the number of \mbox{$2$-convex} objects considered.