On the Reflexivity of Point Sets

Esther M. Arkin, Sándor P. Fekete, Ferran Hurtado, Joseph S. B. Mitchell, Marc Noy, Vera Sacristán, and Saurabh Sethia

Submitted for publication

generalized convexity, angles, simple planar polygons, cover problems, approximation.

We introduce a new measure for planar point sets S. Intuitively, it describes the combinatorial distance from a convex set: The reflexivity ρ(S) of S is given by the smallest number of reflex vertices in a simple polygonization of S. We prove various combinatorial bounds and provide efficient algorithms to compute reflexivity, both exactly (in special cases) and approximately (in general). Our study naturally takes us into the examination of some closely related quantities, such as the convex cover number κ₁(S) of a planar point set, which is the smallest number of convex chains that cover S, and the convex partition number κ₂(S), which is given by the smallest number of disjoint convex chains that cover S. We prove that it is NP-complete to determine the convex cover or the convex partition number, and we give logarithmic-approximation algorithms for determining each.

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