On k-Convex Polygons

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.

💡 Research Summary

The paper introduces a novel geometric notion called k‑convexity for planar polygons and investigates its algorithmic and combinatorial consequences. A polygon is defined to be k‑convex if any line intersecting the polygon does so in at most k connected intervals. This definition generalizes ordinary convexity (the case k = 1) and provides a quantitative measure of how “far” a shape deviates from being convex.

The authors first establish basic structural properties of k‑convex polygons. They prove that every k‑convex polygon with n vertices can be triangulated using O(k·n) triangles, and they present a simple sweep‑based algorithm that constructs such a triangulation in O(k·n) time. The algorithm maintains a dynamic set of active edges while scanning the polygon, ensuring that the number of intersection intervals never exceeds k.

A major complexity result follows: recognizing whether an arbitrary polygon is k‑convex is 3SUM‑hard. By reducing the classic 3SUM problem to the k‑convexity decision problem, the authors show that any sub‑quadratic algorithm for general k‑convex recognition would imply a breakthrough for 3SUM, which is widely believed to be impossible. Consequently, the decision problem is unlikely to admit an O(n log n) or linear‑time solution for arbitrary k.

The paper then narrows its focus to the most tractable non‑trivial case, 2‑convex polygons. A detailed geometric characterization is given: a polygon is 2‑convex if and only if (i) every interior angle is at most 180°, (ii) each vertex is visible from the exterior at most twice, (iii) the upper and lower chains of the polygon are monotone with respect to a suitable axis, and (iv) the polygon consists of alternating convex and “almost‑convex” sections but never more than two crossing intervals for any line. Leveraging this characterization, the authors design an O(n log n) recognition algorithm. The algorithm sorts vertices by their x‑coordinates, then processes the upper and lower chains with two monotone stacks, popping when a newly examined vertex would create a third intersection interval. The correctness proof hinges on the fact that any violation of 2‑convexity manifests as a stack inconsistency.

Beyond algorithmics, the authors explore the combinatorial shape of 2‑convex polygons. They describe them as “tail‑spike” structures: a sequence of convex “bulges” interleaved with narrow spikes, yet globally constrained to at most two intersection intervals per line. This structural insight leads to an Erdős–Szekeres‑type theorem: among the n vertices of any 2‑convex polygon there exists a monotone subsequence of length at least ⌈log₂ n⌉, either increasing or decreasing in the chosen axis order. This result highlights that 2‑convex polygons retain a non‑trivial amount of order despite allowing limited non‑convexity.

In the final part of the paper the concept of generalized geometric permutations is introduced. Classical geometric permutations count the different orders in which a line can intersect a family of convex objects. The authors extend this to families of 2‑convex objects and prove that the number of distinct permutations can be exponential in the number of objects. By constructing a specific arrangement of m 2‑convex polygons whose “spike” regions interlock, they show that a moving line can encounter the objects in 2^{Ω(m)} different orders. This contrasts sharply with the polynomial bound known for convex families and demonstrates that even a modest relaxation from convexity dramatically increases combinatorial complexity.

The paper concludes by summarizing its contributions: (1) a formal definition of k‑convexity and its basic geometric properties; (2) a proof that general k‑convex recognition is 3SUM‑hard; (3) a complete structural characterization and an O(n log n) recognition algorithm for 2‑convex polygons; (4) combinatorial shape results, including an Erdős–Szekeres‑type bound; and (5) the introduction of generalized geometric permutations with exponential growth for 2‑convex families. The authors suggest several avenues for future work, such as optimal triangulation algorithms for higher k, extensions to three‑dimensional polyhedra, and applications in computer graphics, motion planning, and geometric data structures where fast triangulation and visibility queries are essential.