On the convex hull of the integer points in a Bi-circular region

Abstract

Given a set S ⊆ ℝ2, denote Sℤ = S ∩ ℤ2. We obtain bounds for the number of vertices of the convex hull of SZ, where S ⊆ ℝ2 is a convex region bounded by two circular arcs. Two of the bounds are tight bounds-in terms of arc length and in terms of the width of the region and the radii of the circles, respectively. Moreover, an upper bound is given in terms of a new notion of "set oblongness." The results complement the well-known O(r 2/3) bound [2] which applies to a disc of radius r. © Springer-Verlag Berlin Heidelberg 2009.